ON THE UTILITY OF ROBINSON–AMITSUR ULTRAFILTERS
PASHA ZUSMANOVICH

arXiv:0911.5414v8 [math.RA] 20 Aug 2016

A BSTRACT. An embedding theorem for algebraic systems is presented, basing on a certain old ultrafilter construction. As an application, we outline alternative proofs of some results from the theory of PI
algebras, and establish some properties of Tarski’s monsters.

I NTRODUCTION
In 1960s, A. Robinson and S. Amitsur established a number of embedding results in Ring Theory
which proved to be useful in various structural questions. A typical example: if a prime ring R embeds
in a direct product of associative division rings, then R embeds in an associative division ring (see [Am1,
Proof of Theorem 15] and [Am2, Theorem 3] for original papers, and [E, §6] for a nice overview).
The proof of these results follows the same scheme: basing on the initial data – a ring embedded
in a direct product of rings – a certain sort of ultrafilter, which we call a Robinson–Amitsur ultrafilter,
is constructed. Using this ultrafilter, one passes from the direct product of rings to their ultraproduct,
and appeal to the Łoś theorem about elementary equivalence of an algebraic system and its ultraproduct
completes the proof.
In this paper we extend this argument to a class of general algebraic systems (Theorem 1.1), and
observe similarity with the celebrated Jónsson lemma from the universal algebra. Coupled with the
classical Birkhoff theorem about varieties of algebraic systems, this gives a simple yet elegant criterion
for an algebra or group not to satisfy a nontrivial identity (Corollaries 2.5 and 3.2). The corresponding
group result is not entirely new (see comments after Corollary 3.2), but, we believe, its proof is, and the
links between apriori unrelated concepts, ideas and results is the main novelty of this paper.
As an application, we outline alternative, “by abstract nonsense”, proofs of some particular cases
of well-known results from the theory of PI algebras (§4), of results about algebras having the same
identities (§5), and establish that Tarski’s monsters without identities have infinite (relative) girth (§6).
Finally, in §7 we formulate and prove a “dual” version of Theorem 1.1, with embeddings replaced by
surjective homomorphisms, and discuss some its consequences and related questions.
The narrative is occasionally interspersed with questions and speculations.
1. A LGEBRAIC

SYSTEMS

In what follows, by an algebra or ring, we mean an arbitrary, not necessarily associative, or Lie, or
satisfying any other distinguished identities, algebra or ring, unless it is stated otherwise. Algebras are
considered over fields. Ideal of an algebra means a two-sided ideal. By an (algebraic) system we mean
an algebra in the universal algebraic sense, i.e. a set with a number of operations on it of, generally,
various arity.
We deal with algebraic systems whose congruences behave “good enough”, like ideals in rings, or
normal subgroups in groups (or, more generally, ideals in the so-called Ω-groups introduced by P.J. Higgins and studied by Kurosh and his school (see [Ku, Chapter 3, §2]). Namely, suppose the signature Ω of a class of algebraic systems has a 0-ary operation e (i.e., a distinguished element). A term
t(x1 , . . ., xn , y1 , . . . , ym ) composed of the operations in Ω is called an ideal term in y1 , . . ., ym if m > 0 and
t(a1, . . . , an , e, . . ., e) = e
Date: last (cosmetical) revision May 31, 2016.
J. Algebra 388 (2013), 268–286; arXiv:0911.5414.
1

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PASHA ZUSMANOVICH

for any a1 , . . ., an ∈ A for any system A in the class. A subset I of an algebraic system A is called an
ideal if for any ideal term t(x1, . . . , xn , y1 , . . ., ym ) in y1 , . . . , ym , the element
t(a1, . . . , an , b1 , . . ., bm )

(1)

belongs to I for any a1 , . . . , an ∈ A and b1 , . . . , bm ∈ I.
For every congruence θ on A, its equivalence class {a ∈ A | a ≡θ e} is an ideal of A. Generally,
this relation between ideal and congruences is not one-to-one, but if it is, the corresponding class of
algebraic systems is called ideal-determined. In particular, for systems from an ideal-determined class
we can speak about quotients by ideals instead of quotients by congruences.
This notion, along with its numerous particular cases and variations, was studied by Agliano, Chajda,
Fichtner, Grätzer, Gumm, Słomiński, Ursini and others (see, for example, [CEL, Chapter 10]).
Any algebraic system A from an ideal-determined class has at least two ideals – a trivial ideal {e}
and the whole A. The intersection of any two ideals of A is an ideal. A is called finitely subdirectly
irreducible if intersection of any two its nontrivial ideals is nontrivial. An ideal generated by the subset
X of A is the minimal ideal of A containing X , and it coincides with the set of all elements of the form
(1), where t(x1 , . . . , xn , y1 , . . . , ym ) is an ideal term in y1 , . . . , ym , and a1 , . . ., an ∈ A, b1 , . . . , bm ∈ X .
Recall the construction of the ultraproduct – not in the most general form, but in the form suitable for
our purposes. Let {Ai }i∈I be a set of algebraic systems from an ideal-determined class, and F a filter
on the indexing set I. Then


I ∏ Ai , F = { f ∈ ∏ Ai | {i ∈ I | f (i) = e} ∈ F }
i∈I

i∈I

is an ideal of the direct product ∏i∈I Ai , and the quotient by this ideal is called a filtered product of the
set {Ai }i∈I with respect to the filter F , and is denoted by ∏F Ai . In the particular case where all Ai ’s
are isomorphic to the same algebraic system A, their filtered product is called a filtered power of A and
is denoted by AF . When F is an ultrafilter, we speak about ultraproducts and ultrapowers.
An alternative view on ultrafilters and ultraproducts is an analytic one: ultrafilter is a probability
measure taking only two values – 0 and 1, and such that every subset of the indexing set is measurable.
An ultrapower is the collection of measurable functions from the indexing set considered up to the
measure.
We refer to [CEL], [C], [FM], [Ku], or [M] for all necessary basic notions and results related to
universal algebra, and to [BS], [C], [E], or [M] again for ultrafilters and ultraproducts.
Theorem 1.1 (Robinson–Amitsur for algebraic systems). Let {Bi }i∈I be a set of algebraic systems from
an ideal-determined class. If a finitely subdirectly irreducible algebraic system A embeds in the direct
product ∏i∈I Bi , then there is an ultrafilter U on the set I such that A embeds in the ultraproduct ∏U Bi .
Proof. Define
S = {{i ∈ I | f (i) 6= e} | f ∈ A, f 6= e}.
Let us verify that intersection of any two elements of S contains an element of S . Let S, T ∈ S , say,
S = {i ∈ I | f (i) 6= e} and T = {i ∈ I | g(i) 6= e} for some f , g ∈ A different from e. Since A is finitely
subdirectly irreducible, it contains an element u 6= e belonging to the intersection of ideals generated by
{ f } and {g}. Let i ∈ I such that f (i) = e. Since u = t(h1, . . . , hn , f , . . ., f ) for some ideal term t and
h1 , . . ., hn ∈ A,
u(i) = t(h1, . . ., hn , f , . . . , f )(i) = t(h1(i), . . ., hn (i), f (i), . . ., f (i))
= t(h1(i), . . ., hn (i), e, . . ., e) = e.
Coupling this with a similar assertion for g, we get that
S ∩ T ⊃ {i ∈ I | u(i) 6= e} ∈ S .
Thus S satisfies the finite intersection property and is contained in some ultrafilter U on I. Factoring
the embedding of algebraic systems A ֒→ ∏i∈I Bi by the ideal I(∏i∈I Bi , U ), we get an embedding of

ON THE UTILITY OF ROBINSON–AMITSUR ULTRAFILTERS

algebraic systems
A

.

A∩I



∏ Bi , U
i∈I



֒→



∏ Bi
i∈I

3


. 
I ∏ Bi , U = ∏ Bi .
i∈I

U


Let f ∈ A ∩ I ∏i∈I Bi , U . Then {i ∈ I | f (i) = e} ∈ U , and, since U is an ultrafilter, {i ∈ I | f (i) 6=
e} ∈
/ U , and hence {i∈ I | f (i) 6=e} ∈
/ S . From f ∈ A and the definition of S it follows that f = e.
This shows that A ∩ I ∏i∈I Bi , U = {e}.



The ultraproduct construction used in this proof mimics the old one, used by A. Robinson and S.
Amitsur in Ring Theory, mentioned in the introduction.
The finite subdirect irreducibility of an algebraic system A is equivalent to the following condition:
if A embeds in the finite direct product of algebraic systems ∏ni=1 Bi , then A embeds in one of Bi ’s.
An infinite analog of this condition is subdirect irreducibility, that is, the condition that intersection of
any (possibly infinite) set of nontrivial ideals of A is nontrivial (or, equivalently, A possesses a minimal
nontrivial ideal which is called monolith). Similarly, the latter condition is equivalent to the following: if
A embeds in the (possibly infinite) direct product ∏i∈I Bi , then A embeds in one of Bi ’s. Thus, Theorem
1.1 can be considered as, perhaps, somewhat surprising statement that for ideal-determined classes,
finite subdirect irreducibility implies a sort of a weaker form of subdirect irreducibility.
An application of Theorem 1.1 to varieties and quasivarieties of algebraic systems follows.
If A is an algebraic system, Var(A) and Qvar(A) denote, respectively, a variety and a quasivariety
generated by A. Any quasivariety (and, in particular, any variety) possesses free algebraic systems (this
is formulated explicitly, for example, in [C, Chapter VI, Proposition 4.5] and is implicit in [M, Chapter
V]).
Corollary 1.2. Let A be an algebraic system from an ideal-determined class. A finitely subdirectly
irreducible free system in Var(A) or Qvar(A) embeds in an ultrapower of A.
Proof. Let F be a free system in Var(A) which is finitely subdirectly irreducible. According to the
Birkhoff theorem, F = B/I for an ideal I of an algebra B, and B is a subalgebra of a direct power of A.
Because of the universal property of F , the short exact sequence {e} → I → B → F → {e} splits, i.e.
F embeds in B, and hence in a direct power of A. Then apply Theorem 1.1.
Similarly, according to the Birkhoff-like characterization of quasivarieties due to Malcev ([M, Chapter
V, §11, Theorem 4]), if F is a free system in Qvar(A), then it is a subalgebra of a filtered power
AF = AI /I(AI , F ) of A. Taking preimage of F with respect to the homomorphism AI → AI /I(AI , F ),
we get that F is a quotient of a subalgebra in AI , and the rest of reasoning is the same as above.

Compare Theorem 1.1 and Corollary 1.2 with the celebrated Jónsson lemma in the generalized form
due to combined efforts of Freese, Hagemann, Herrmann, Hrushovski and McKenzie (see [FM, Theorem 10.1]): if A is a subdirectly irreducible algebraic system from a modular variety (i.e., the congruence
lattice of any system from the variety is modular) generated by a set {Bi }i∈I of algebraic systems, then
the quotient of A by the centralizer of its monolith embeds in a homomorphic image of a subsystem of an
ultraproduct ∏U Bi . Note that the congruence (=ideal) lattice of any algebra from an ideal-determined
class is modular (see, for example, [CEL, Remark 10.1.16]).
Corollary 1.3 (Criterion for absence of non-trivial identities for algebraic systems). Let V be a variety
of algebraic systems from an ideal-determined class, and suppose that all free systems of V are finitely
subdirectly irreducible. Then for an algebraic system A ∈ V, the following is equivalent:
(i) any identity of A is an identity of V (i.e., A does not satisfy nontrivial identities within V);
(ii) any free system of V embeds in an ultrapower of A;
(iii) any free system of V embeds in a system elementarily equivalent to A.
Proof. (i) ⇒ (ii) follows from Corollary 1.2.

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PASHA ZUSMANOVICH

(ii) ⇒ (iii) follows from the Łoś theorem about elementary equivalence of an algebraic system and
its ultrapower.
(iii) ⇒ (i) An algebraic system elementarily equivalent to A does not satisfy a nontrivial identity
within V. Since the latter is the first-order property, A does not satisfy a nontrivial identity either.

Remarks.
(1) Of course, the equivalence of conditions (ii) and (iii) also follows from the (powerful) Keisler
ultrapower theorem (see, for example, [BS, Chapter 7, Corollary 2.7]).
(2) Recall a well-known fact from model theory: an algebraic system B embeds in an ultrapower of
an algebraic system A if and only if T h∀ (A) ⊆ T h∀ (B), where T h∀ denotes the universal theory of a
system (see, for example, [BS, Chapter 9, Lemma 3.8]). This allows to rephrase condition (ii) or (iii) as
follows: the universal theory of A is contained in the universal theory of any free system of V.
(3) As any variety is determined by its free system of countable rank (see, for example, [C, Chapter
IV, Proposition 3.8] or [M, Chapter VI, §13, Theorem 3]), in conditions (ii) and (iii) of the corollary, as
well as in (2) above, one may replace “any free system” by “the free system of countable rank”.
2. A LGEBRAS
Specializing results of the previous section to the (ideal-determined, obviously) class of rings, we
get, for example, a nonassociative analog of one of the classical embedding results in Ring Theory
mentioned in the introduction:
Corollary 2.1. If a finitely subdirectly irreducible ring A embeds in a direct product of division rings,
then A embeds in a division ring.
Proof. By Theorem 1.1, A embeds in an ultraproduct of division rings. As the property to be a division
ring is the first-order property, by the Łoś theorem the ultraproduct of division rings is a division ring,
whence the conclusion.

When trying to specialize to the class of algebras, we should deal with the issue of the base field.
For example, Corollary 1.3 is not applicable directly. The problem is this: Corollary 1.3 is obtained by
combination of the Birkhoff theorem about varieties of algebras (or a similar statements), and the Łoś
theorem about elementary equivalence of an algebra and its ultraproduct. One may treat algebras either
as two-sorted theories (algebra over a field, field), or distinguish elements of the base field by an unary
predicate. Either way, while in Birkhoff-like reasonings the base field remains the same, in Łoś ones it,
generally, changes: we pass to an ultraproduct of the base field.
So, for now on, fix the base field K. The embedding claimed in Theorem 1.1 is an embedding of
algebras defined over K. Of course, the ultraproduct ∏U Bi is defined also over the ultrapower field
K U , so we have an embedding of K U A in ∏U Bi as K U -algebras. (Here K U A is understood as a K U linear span of A, considered as a K-subalgebra of ∏U Bi ). Due to the universal property of the tensor
product, there is a surjection of K U -algebras
(2)

A ⊗K K U → K U A,

but this surjection is, generally, not a bijection.
An important observation is that for free algebras A in the major varieties of algebras considered in
the literature – the varieties of all algebras, associative algebras, and Lie algebras, the surjection (2) is a
bijection.
Lemma 2.2. Let A be a subalgebra of an algebra B, both defined over a field K. Suppose B is also
defined over a field F containing K, and that as a K-algebra, A does not have commutative subspaces
(i.e., subspaces all whose elements commute) of dimension > 1. Then FA ≃ A ⊗K F as F-algebras.
Proof. The claimed isomorphism follows from the fact that the linear dependence of elements of A over
F implies their linear dependence over K. Indeed, consider the linear dependence
(3)

f1 a1 + · · · + fn an = 0,

ON THE UTILITY OF ROBINSON–AMITSUR ULTRAFILTERS

5

where fi ∈ F, ai ∈ A, and let us prove by induction on n that a1 , . . ., an are linearly dependent over K.
For n = 1 this is trivial. Taking the commutator with an of both sides of (3), we get
f1 [a1 , an ] + · · · + fn−1 [an−1 , an ] = 0.
(we use the usual notation for the commutator: [a, b] = ab − ba). By inductive hypothesis, there are
k1 , . . . , kn−1 ∈ K, not all zero, such that
k1 [a1 , an ] + · · · + kn−1 [an−1 , an ] = [k1 a1 + · · · + kn−1 an−1 , an ] = 0.
But since A does not have commutative subspaces of dimension > 1,
k(k1 a1 + · · · + kn−1 an−1 ) + ℓan = 0
for some k, ℓ ∈ K, not both zero.



Obviously, the hypothesis of this lemma is satisfied when A is a free algebra or a free Lie algebra of
rank > 1 (in view of the Shirshov–Witt theorem). A similar, but just a little bit more involved argument
(see, for example, proof of Lemma 1 in [MM]) shows that the conclusion of the lemma holds also when
A is a free associative algebra of rank > 1.
Another issue we should deal with when trying to apply results of the previous section to the class of
algebras, is when the hypothesis of Corollary 1.2 holds, i.e., when free algebras in a variety are finitely
subdirectly irreducible. Again, this hypothesis is satisfied for absolutely free algebras, free associative
algebras, and free Lie algebras of rank > 1 (on the other hand, it is not satisfied for free Jordan algebras
and free alternative algebras). In fact, those free algebras satisfy a stronger condition – primeness,
and, in the context of algebras, we will mainly focus on the latter condition instead of finite subdirect
irreducibility.
Let V be a variety of algebras, and F (X ) is the free algebra in this variety generated by a set X . By
words we mean elements of F (X ) for some X . The standard grading on F (X ) is defined by length of
words. By 2-nontrivial words we mean words degrees of all whose homogeneous components in each
of the first two indeterminates are non zero. For example,
xy,

(xy)x − (xy)(xz),

(zy)x + 2(tx)y + x(yz)(tx)

are 2-nontrivial words in x, y, z,t (in that order), while
x,

xz,

(xy)x + xz,

(yz)(yt) + (xy)z

are not.
Lemma 2.3. For an algebra A ∈ V, the following is equivalent:
(i) For any two nonzero ideals I, J of A, IJ 6= 0.
(ii) For any two nonzero elements x, y ∈ A, there is a 2-nontrivial word w(ξ1 , . . . , ξn ), n ≥ 2 and
elements x1 , . . ., xn−2 ∈ A such that w(x, y, x1 , . . . , xn−2 ) 6= 0.
Proof. (i) ⇒ (ii). Suppose there are nonzero x, y ∈ A such that for any 2-nontrivial word w(ξ1 , . . ., ξn )
and any x1 , . . . , xn−2 ∈ A, w(x, y, x1 , . . . , xn−2 ) = 0. Let I and J be ideals of A generated by x and y
respectively. Clearly IJ = 0, a contradiction.
(ii) ⇒ (i). Suppose I, J are two nonzero ideals of A. Taking x ∈ I and y ∈ J, we have w(x, y, x1 , . . . , xn−2 ) 6=
0 for some 2-nontrivial word w and elements xi of A. Clearly,
w(x, y, x1 , . . ., xn−2 ) ∈ I ∩ J.
Further, there is a 2-nontrivial word u and elements y1 , . . . , ym−2 ∈ A such that
u(w(x, y, x1, . . . , xn−2 ), w(x, y, x1 , . . ., xn−2 ), y1 , . . . , ym−2 ) 6= 0.
In particular, for some monomial m occurring in u, we have
(4)

m(w(x, y, x1 , . . ., xn−2 ), w(x, y, x1 , . . ., xn−2 ), y1 , . . . , ym−2 ) 6= 0.

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PASHA ZUSMANOVICH

Examining how m is built up from the variables, there must be some point where a sub-monomial
containing the first variable is multiplied by a sub-monomial containing the second variable, giving a
product pq where p involves one of the first two variables, and q the other.
Due to (4),
p(w(x, y, x1 , . . ., xn−2 ), w(x, y, x1 , . . . , xn−2 ), y1 , . . . , ym−2 )
× q(w(x, y, x1 , . . ., xn−2 ), w(x, y, x1 , . . . , xn−2 ), y1 , . . ., ym−2 ) 6= 0.
Each of these factors belongs to I ∩ J because w(x, y, x1 , . . . , xn−2 ) does, so their product belongs to
(I ∩ J)(I ∩ J) ⊆ IJ, as required.

An algebra A ∈ V satisfying the equivalent conditions of Lemma 2.3, is called V-prime. When V is
a variety of all associative algebras, this notion coincides with the classical notion of a prime associative
algebra.
Clearly, if W is another variety and V ⊆ W, an algebra A ∈ V is V-prime if and only if it is W-prime,
so we can simply speak about prime algebras (which are prime in the variety of all algebras).
As for any two ideals I and J, IJ ⊆ I ∩ J, prime algebras (or rings) are finitely subdirectly irreducible.
Now, the claim about primeness of free algebras and free associative algebras is obvious, as they do
not have zero divisors. To establish primeness of a free Lie algebra of rank > 1, consider two nonzero
ideals I, J in it. By the Shirshov–Witt theorem about freeness of subalgebras of a free Lie algebra,
neither of I, J can be one-dimensional, and hence we may choose two linearly independent elements
x ∈ I and y ∈ J. Their commutator is nonzero, as otherwise they would form a 2-dimensional abelian
subalgebra, which again contradicts the Shirshov–Witt theorem. Hence [I, J] 6= 0.
Another situation when relatively free algebras are prime is described by the following lemma.
Lemma 2.4. Let A be a prime algebra over an infinite field. Then the free algebra of infinite rank in
Var(A) is prime.
Proof. Let F (X ) be the free algebra in Var(A) freely generated by an infinite set X = {x1 , x2 , . . . }.
Suppose that there are nonzero elements u(x1 , . . ., xn ), v(x1 , . . . , xn ) ∈ F (X ) such that
(5)

w(u, v, u1 , . . ., um ) = 0

for any 2-nontrivial word w and u1 , . . ., um ∈ F (X ). As any relation between free generators of F (X ) is
an identity in A, the equalities (5) are identities in A. Taking u1 = xn+1 , u2 = xn+2 , . . . , we get that
w(u(x1 , . . . , xn ), v(x1 , . . . , xn ), xn+1 , . . ., xn+m ) = 0
for any 2-nontrivial word w and any x1 , . . . , xn , xn+1 , . . . , xn+m ∈ A. As A is prime, either
u(x1 , . . . , xn ) = 0 or v(x1 , . . . , xn ) = 0 for any x1 , . . . , xn ∈ A.
Fix some basis of A, take (x1 , . . . , xn ) ∈ An = A × · · · × A (n times) at which u is nonzero, (y1 , . . . , yn ) ∈
n
A at which v is nonzero, and consider the line in An of all affine linear combinations
(6)

λ (x1 , . . . , xn ) + (1 − λ )(y1 , . . ., yn ),

where λ is an element of the base field K. On this line, the coordinates of u and v in terms of the chosen
basis are each given by a polynomial in λ . So, picking a K-linear function on A such whose value at
u(x1 , . . . , xn ) is nonzero and another such whose value at v(y1 , . . . , yn ) is nonzero, we see that for all but
finitely many λ ∈ K, both u and v will be nonzero on the n-tuple (6), a contradiction.
If A is finite-dimensional, we can use a similar, but more elegant Zariski-topology argument instead:
both sets of n-tuples of elements of A on which u, respectively v, does not vanish, form a nonempty
Zariski-open subset in An , whence they have a nonzero intersection, a contradiction.

All these observations lead to a variant of the general Corollary 1.3 for the case of algebras:
Corollary 2.5 (Criterion for absence of non-trivial identities for algebras). For an algebra A belonging
to one of the following varieties of algebras: all algebras, associative algebras, or Lie algebras, the
following are equivalent:

ON THE UTILITY OF ROBINSON–AMITSUR ULTRAFILTERS

7

(i) A does not satisfy a nontrivial identity;
(ii) any free algebra embeds in an ultrapower of A;
(iii) any free algebra embeds in an algebra elementarily equivalent to A.
Proof. (i) ⇒ (ii) follows from Corollary 1.2. Note that in view of Lemma 2.2 (applied to the case F =
K U , where K is the base field, and U is an appropriate ultrafilter) and remarks after it, the embedding
of Corollary 1.2 can be considered as an embedding of K U -algebras.
(ii) ⇒ (iii) as in the proof of Corollary 1.3: by the Łoś theorem, the pairs (B, K) and (BU , K U ) are
elementarily equivalent as models of the two-sorted theory (algebra over a field, field).
(iii) ⇒ (i) as in the proof of Corollary 1.3.

All the remarks after Corollary 1.3 are applicable. In addition, as the free associative, respectively
Lie, algebra of countable rank embeds in the free associative, respectively Lie, algebra of rank 2, for
these varieties one can replace “any free algebra” in conditions (ii) and (iii) by “the free algebra of rank
2”.
Let us provide some negative examples showing that conclusions of some statements of this section
do not always hold.
The conclusion of Lemma 2.2 does not hold for polynomial algebras in > 1 variables (i.e., for free
associative commutative algebras). Indeed, let F = K(x) (the field of rational functions in 1 variable)
and A = K[x, y] (the algebra of polynomials in 2 variables), considered as a K-subalgebra of K(x, y) (the
field of rational functions in 2 variables). Then
FA = K(x)[y] ≃ K[y] ⊗K K(x),
but A ⊗K F = K[x, y] ⊗K K(x).
In some other situations, the surjection (2) also can be very far from being a bijection. For example,
consider the free algebra F of countable rank in the variety Var(A) generated by a finite-dimensional
prime algebra over an infinite field K. By Lemma 2.4, F is prime, and by Corollary 1.2, F embeds,
as a K-algebra, in an ultrapower AU ≃ A ⊗K K U (see Corollary 4.3 below). Hence K U F is a finitedimensional K U -algebra. On the other hand, since F is residually nilpotent (see, for example, [Ba,
§4.2.10] for the case of Lie algebras; the general case is treated identically), the finite-dimensionality
of F would imply nilpotency of F , and hence nilpotency of A, a contradiction. Consequently, F is
infinite-dimensional over K, and F ⊗K K U is infinite-dimensional over K U . We will use essentially the
same type of arguments below in §5, when describing an alternative approach to a result of Kushkulei
and Razmyslov about varieties generated by simple finite-dimensional Lie algebras.
3. G ROUPS
As the class of (all) groups is, obviously, ideal-determined, Theorem 1.1 and Corollary 1.2 apply to
it.
Let us single out an important condition for groups of which the finite subdirect irreducibility is a
consequence:
Lemma 3.1. A group in which any two commuting elements generate a cyclic subgroup either of prime
order, or of infinite order, is finitely subdirectly irreducible.
Proof. Let G be a group with the given property, and N, M be two normal subgroups of G. Take
x ∈ N, y ∈ M, x, y 6= 1. If x, y do not commute, then 1 6= xyx−1 y−1 ∈ N ∩ M. If x, y commute, then
they generate a cyclic subgroup of G, generated by a single element a ∈ G, either of prime order p,
or of infinite order. Write x = an , y = am for some 0 < n, m < p in the first case, and some nonzero
integers n, m in the second one. We have anm ∈ M ∩ N, and due to the restriction on n, m, anm 6= 1 in
both cases.

Now we can establish Corollary 1.3 for the variety of all groups and for Burnside varieties:

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PASHA ZUSMANOVICH

Corollary 3.2 (Criterion for absence of non-trivial identities for groups). For a group G belonging to
one of the following varieties: all groups, groups satisfying the identity x p = 1 for a prime p ≥ 673, the
following are equivalent:
(i) G does not satisfy a nontrivial identity within the given variety;
(ii) any free group in the variety embeds in an ultrapower of G;
(iii) any free group in the variety embeds in a group elementarily equivalent to G.
The restriction on p is stipulated, of course, by the celebrated Novikov–Adian solution of the Burnside
problem (see [Ad]): the Burnside group B(n, m) of exponent m freely generated by n elements, is infinite
for odd m ≥ 665, and 673 is the next prime after 665.
Proof. Follows from Corollary 1.3, Lemma 3.1, and the fact that the corresponding free groups satisfy
the hypothesis of Lemma 3.1: from [Ad, Chapter VI, §3, Theorem 3.3] it follows that under the given
restriction on p, any abelian subgroup of the Burnside group B(n, p) is cyclic of order p, and from
the Nielsen–Schreier theorem about freeness of subgroups of free groups it follows that any abelian
subgroup of an absolutely free group is infinite cyclic.

The same remarks as those after Corollary 2.5 apply also in the group case. In particular, condition
(ii) is equivalent to the condition that the universal theory of G is contained in the universal theory of
any free group. As in the case of Lie and associative algebras, “any free group” in conditions (ii) and
(iii) of the corollary, can be replaced by “the free group of rank 2”: an embedding of the free group of
countable rank into the free group of rank 2 is well-known in the case of absolutely free groups, and
is established in [Š] (see also [At, §3] and references therein for the later alternative approaches) in the
case of Burnside groups.
Corollary 3.2 is probably known to experts – at least statements equivalent to the implication (i) ⇒
(ii) in the case of the variety of all groups can be found in [Bo, Theorem 1] and [DS, Lemma 6.15]. The
proofs there are different and based on the fact that ultraproducts are ω1 -compact.
Question. Is there a semigroup property such that the corresponding analogs of the results of this section would hold in the class of all semigroups? The same question for the classes of inverse semigroups
and quasigroups.
Note that neither of these classes is ideal-determined.
4. A PPLICATION : PI

ALGEBRAS

As an application of this machinery, let us demonstrate how one can handle, in a way different from
the traditional approaches, some well-known statements from the theory of associative algebras satisfying polynomial identities (usually called PI algebras).
The Regev celebrated “A ⊗ B” theorem asserts that the tensor product of two PI algebras A and B is PI
(see, for example, [KR, Theorem 5.42]). If we want to prove it using results of §1, we encounter a few
difficulties: first, to establish relationship between the ultrapower of the tensor product (A ⊗B)U and the
tensor product of ultrapowers AU ⊗ BU (perhaps, considering some sort of completed tensor product
instead of the usual one may help), and, second, to be able to say something about algebras A and B such
that their (possibly completed) tensor product contains a free associative algebra. But at least, in this
way we are able to provide an alternative proof of the particular case where one of the tensor factors is
finite-dimensional (first established by Procesi and Small in [PS] for the even more particular case where
one of the tensor factors is a full matrix algebra; a quantitative refinement (degrees of the corresponding
standard identities) of Procesi–Small result was later given in [Do, Theorem 3.2]). This particular case is
morally important, as semiprime PI algebras embed in matrix algebras over commutative rings (see, for
example, [KR, Remark 1.69]), which essentially reduces the semiprime situation to a finite-dimensional
one.
Theorem 4.1 (“Baby Regev’s A ⊗ B”). The tensor product of two associative algebras, one of them PI
and the other finite-dimensional, is PI.

ON THE UTILITY OF ROBINSON–AMITSUR ULTRAFILTERS

9

Lemma 4.2. Let A, B be algebras defined over a field K, A finite-dimensional. Then, for any ultrafilter
U,
(A ⊗K B)U ≃ A ⊗K BU
(as K-algebras).
Special cases of this assertion were proved many times in literature – for example, in [NN] for the
case where A is an associative full matrix algebra, and in [T, Proposition 25] for the case where both A
and B are finite-dimensional, and the proof is standard.
Proof. Let {a1 , . . ., an } be a basis of A. Obviously, for each K-algebra C, each element of A ⊗K C can
be uniquely represented as ∑nk=1 ak ⊗ ck for some ck ∈ C. Define a map

ϕ : (A ⊗K B)I → A ⊗K BI
as follows: for f ∈ (A⊗K B)I write f (i) = ∑nk=1 ak ⊗bki , i ∈ I and define ϕ ( f ) ∈ A⊗K BI as ∑nk=1 ak ⊗gk ,
where gk ∈ BI is defined as gk (i) = bki , i ∈ I. Writing multiplication in A in terms of the basis elements,
one can see that ϕ is an isomorphism of K-algebras.
The ideal I((A ⊗K B)I , U ) maps under ϕ to A ⊗K I(BI , U ), so factoring out both sides of the isomorphism ϕ by the corresponding ideals, we get:
(A ⊗K B)U = (A ⊗K B)I /I((A ⊗K B)I , U ) ≃ (A ⊗K BI )/(A ⊗K I(BI , U ))
≃ A ⊗K (BI /I(BI , U )) = A ⊗K BU .

Corollary 4.3 ([T, Proposition 21]). Let A be a finite-dimensional algebra defined over a field K. Then,
for any ultrafilter U , AU ≃ A ⊗K K U (as K-algebras).
Proof. Put B = K.



Proof of Theorem 4.1. Let A be a finite-dimensional associative algebra, and B a PI algebra, defined over
a field K. Suppose A ⊗K B is not PI. Then by Corollary 2.5, some ultrapower (A ⊗K B)U , considered as
a K U -algebra, contains a free associative subalgebra F of finite rank. By Lemma 4.2,
(A ⊗K B)U ≃ (A ⊗K K U ) ⊗K U BU
as K U -algebras.
Since F is finitely-generated, we may choose a finitely-generated K U -subalgebra B′ of BU such that
F is a subalgebra of (A ⊗K K U ) ⊗K U B′ . Since B is PI, BU is PI, and B′ is PI. The Shirshov height
theorem implies that B′ has polynomial growth (or, in other words, its Gelfand-Kirillov dimension is
finite; see, for example, proof of Theorem 9.19 in [KR]). As A ⊗K K U is finite-dimensional (over K U ),
the tensor product (A ⊗K K U ) ⊗K U B′ has polynomial growth too. But this contradicts the fact that its
subalgebra F has exponential growth.

Needless to say, this proof, unlike those in [PS], as well as all the proofs of the full-fledged Regev’s
A ⊗ B, is absolutely non-constructive, as it uses existence of an ultrafilter, and, therefore, axiom of
choice.
Along the same lines one may treat, at least in some particular cases, a number of other well-known
results from PI theory: commutativity of an ordered PI algebra; PIness of a finitely-graded algebra
with PI “null component”; of an algebra with a group action whose fixed point subalgebra is PI; of a
localization of a PI algebra; of algebras with involution, etc.
Let us note yet another immediate application to a situation which resonates with the universalalgebraic setup of §1. A number of authors considered a property of definable principal congruences in
algebraic systems. For rings, this amounts to saying that the property that an element x of a ring belongs
to the (principal) ideal generated by an element y, is expressed as a first-order formula in free variables
x, y of the theory of rings. For an associative ring R, the fact that Var(R) has definable principal congruences, can be expressed as a certain formula of the universal theory of rings, evidently not satisfied in

10

PASHA ZUSMANOVICH

free associative rings (see [Si, Lemma 1]). From this and from the ring-theoretic version of Corollary
2.5 immediately follows that if Var(R) has definable principal congruences, then R is PI, what is the
main result of [Si], obtained there with appeal to some deep result from PI theory.
One may try to apply the same reasoning to the known open problem: suppose an associative algebra
R is represented as the vector space sum of its subalgebras: R = A+B. If A, B are PI, is it true that R is PI?
(see [FGL] and [KP] with a transitive closure of references therein). It is easy to see that the operation
of taking ultraproduct commutes with the operation of taking the vector space sum: RU = AU + BU .
Consequently, the question can be reduced to the following one: is it possible that the vector space
sum A + B of two PI algebras can contain a free associative subalgebra? If A + B is finitely-generated,
the impossibility of this follows from the same growth argument as in the proof of Theorem 4.1. The
difficulty, however, lies in the fact that it is not clear how to reduce the situation to a finitely-generated
one, as the multiplication between A and B can be intertwined in a complicated way.
5. A PPLICATION :

ALGEBRAS WITH THE SAME IDENTITIES

In [R, §5] (see also [SZ]), Kushkulei and Razmyslov obtained results stating that some classes of
finite-dimensional algebras (e.g., prime over an algebraically closed field) are uniquely determined by
their identities. Another result in this direction:
Theorem 5.1. Let P be a class of finite-dimensional algebras satisfying the following conditions:
(i) If A, B ∈ P, A and B are defined over the same field, A is a subalgebra of B, and Var(A) =
Var(B), then A = B.
(ii) P is closed under elementary equivalence in the first-order two-sorted theory of pairs (algebra
over a field, field).
(iii) P contains all finite-dimensional prime algebras.
Then P satisfies the following strengthening of condition (i): if A, B ∈ P, A and B are defined over the
same field, and Var(A) = Var(B), then A ≃ B.
We stress that the base field over which algebras in the class P are defined, is not fixed.
Proof. Let A, B ∈ P, both defined over a field K, be such that Var(A) = Var(B). Lemma 2.4, Corollary
1.2 and Corollary 4.3 imply that a free algebra F in the variety Var(A) = Var(B) embeds, as a Kalgebra, into an algebra A ⊗ K U for some ultrafilter U . Hence K U F is isomorphic, as a K U -algebra,
to a subalgebra of A ⊗ K U . It is easy to see that primeness of the K-algebra F implies primeness of the
K U -algebra K U F . By (iii), the latter algebra belong to P, and by the Łoś theorem and (ii), the K U algebra A ⊗K K U belongs to P. Obviously, K U F satisfies over K U the same identities as A over K.
Then by (i), K U F ≃ A ⊗K K U . By the same reasoning, K U F ≃ B ⊗K K U . Hence A ⊗K K U ≃ B ⊗K K U
as K U -algebras, and A ≃ B as K-algebras.

This theorem allows, for example, to obtain an alternative proof of results of Kushkulei and Razmyslov in an important particular case of finite-dimensional simple Lie algebras over an algebraically
closed field of characteristic zero (see [R, §5, Corollary 1 and Comments]). Indeed, the hypothesis of
the theorem is satisfied for such class of algebras: (i) can be proved with the help of the well-known
Dynkin’s classification [Dy] of semisimple subalgebras of semisimple Lie algebras, (ii) is evident, and
(iii) follows from the obvious fact that for finite-dimensional Lie algebras over a field of characteristic
zero, simplicity is equivalent to primeness.
The same approach can be applied to finite-dimensional simple Jordan algebras (which follows from
the general results of Kushkulei and Razmyslov and also established independently in [DR]), as well as
to graded simple associative algebras ([KZ]).
Along the same lines one may treat varieties generated by affine Kac–Moody algebras. Consider, for
example, a Lie algebra of the form
(7)

b
g = (g ⊗K K[t,t −1]) ⊕ Kz,

ON THE UTILITY OF ROBINSON–AMITSUR ULTRAFILTERS

11

where g is a split finite-dimensional simple Lie algebra defined over a field K of characteristic zero,
K[t,t −1] is the algebra of Laurent polynomials, z is the central element, and the multiplication between
elements of g ⊗ K[t,t −1] is twisted by the well-known 2-cocycle:
df
[x ⊗ f , y ⊗ g] = [x, y] ⊗ f g + (x, y)Res( g)z
dt
−1
where x, y ∈ g, f , g ∈ K[t,t ], and ( · , · ) is the Killing form on g.
In [Za] it is shown, among other, that Var(b
g) = [Var(g), E], where E is the variety consisting of
the single zero algebra, and [· , · ] is the standard commutator of varieties as defined in [Ba, §4.3.8]
(in other words, for a variety V, [V, E] is nothing but a variety defined by identities of the form
[ f (x1 , . . . , xn ), xn+1 ] = 0, where f (x1 , . . . , xn ) = 0 is an identity in V). Let us complement this result
by showing that free algebras in Var(b
g) embed in algebras whose structure closely resembles those of
b
g.
By [St, Corollary 2.2], every ideal of the Lie algebra g ⊗ K[t,t −1] is of the form g ⊗ I, where I is an
ideal of K[t,t −1]. Since K[t,t −1] does not have zero divisors, it is prime, hence g ⊗ K[t,t −1] is prime,
and b
g is a central extension of a prime Lie algebra. By an obvious modification of the proof of Lemma
2.4, one get that for the free algebra L in Var(b
g) of countable rank, L/Z(L) is prime, and, by Corollary
−1
U
1.2 and Lemma 4.2, embeds in g ⊗ K[t,t ] for some ultrafilter U . From the results of [Ba, §4.4]
it follows that L embeds in a central extension of g ⊗ K[t,t −1]U . The latter, by [Ka, Theorem 3.3
and Corollary 3.5], is described in terms of the first-order cyclic homology of K[t,t −1]U , so we get an
embedding


L ֒→ g ⊗K U K[t,t −1]U ⊕ HC1 (K[t,t −1]U ).
The multiplication in the right-hand side Lie algebra is defined by the formula
[x ⊗ F, y ⊗ G] = [x, y] ⊗ FG + (x, y)F ∧ G,
where x, y ∈ g, F, G ∈ K[t,t −1]U , and F ∧ G denotes the corresponding homology class in
HC1 (K[t,t −1]U ).
The addition to (7) of the Kt dtd term, or twisting by automorphisms of g, do not significantly change
the picture, and can be treated in the same way.
Similarly, one may treat varieties generated by modular semisimple Lie algebras. According to the
classical Block theorem, a typical finite-dimensional semisimple Lie algebra over a field K of characteristic p which is not isomorphic to the sum of simple ones, has the form

 

p
p
S ⊗K K[t1, . . . ,tn]/(t1 , . . . ,tn ) ⊕ 1 ⊗K D ,
p

p

where S is a simple Lie algebra, K[t1 , . . .,tn ]/(t1 , . . . ,tn ) is the reduced polynomial algebra, and D is a
p
p
derivation algebra of K[t1, . . . ,tn]/(t1 , . . .,tn ) such that the latter does not have D-invariant ideals.
To such algebras, Lemma 2.4 is applicable, and, as in the Kac–Moody case, further application of
Corollary 1.2 and Lemma 4.2 gives an embedding (as K U -algebras)

 

L ֒→ S ⊗K U K U [t1, . . . ,tn]/(t1p, . . . ,tnp) ⊕ 1 ⊗K U (D ⊗K K U )
for some ultrafilter U .
6. A PPLICATION : TARSKI ’ S

MONSTERS

Under Tarski’s monster of type p, p being a prime (respectively, of type ∞) we understand an infinite
nonabelian group all whose proper subgroups are cyclic of order p (respectively, of infinite order).
Such groups were constructed, among other groups with exotic-looking restrictions on subgroups, by
Olshanskii in the framework of his celebrated machinery of geometrically-motivated manipulations with
group presentations (see, for example, [O1, Chapter 9, §28.1]). In Olshanskii’s works, the existence of
Tarski’s monsters of type p is established for p > 1075 . Later, in [AL] this estimate has been reduced to
p > 1003.

12

PASHA ZUSMANOVICH

Let G be a finitely-generated group, and
(8)

{1} → N → F → G → {1}

its presentation, where F is a free group of finite rank, and N is a normal subgroup of relations. The
girth of the presentation (8) is the minimal length of elements of N , i.e., the minimal length of relations
between the chosen generators of G (or, in other words, the minimal length of a simple loop in the
corresponding Cayley graph). The girth of G is the supremum of girths of all its presentations with a
finite number of generators. This natural notion was introduced and studied recently by Akhmedov in
[Ak1] and [Ak2], and by Schleimer in [Sc]. One of the interesting questions arising in that regard is to
construct groups of infinite girth.
As noted in the above-mentioned works, a group satisfying a nontrivial identity cannot have an infinite
girth. To circumvent this obstacle, let us introduce the notion of relative girth – a girth relative to all
identities a group satisfies: in the definition of girth above, replace in (8) the absolutely free group F by
a free group (of finite rank) in the variety Var(G).
Theorem 6.1.
(i) A Tarski’s monster of type p does not satisfy any nontrivial identity except x p = 1 and its consequences, if and only if it has infinite relative girth.
(ii) A Tarski’s monster of type ∞ does not satisfy any nontrivial identity if and only if it has infinite
girth.
Remark (A. Olshanskii). Tarski’s monsters satisfying condition (ii) of the theorem do exist (and, moreover, there is an abundance of them), and they can be constructed in the following way.
According to [O2, Corollary 1], each non-cyclic torsion-free hyperbolic group G0 has a homomorphic
image G which is a Tarski’s monster of type ∞. Such monsters are constructed by subsequent applications of [O2, Theorem 2], as the direct limit of a system of surjective maps of groups G0 → G1 → . . . .
Each Gn is a non-cyclic torsion-free hyperbolic group (and hence contains a free subgroup of countable
rank), and is 2-generated for n ≥ 1. Let us denote, by abuse of notation, the corresponding generators
by the same letters a, b (so, a, b ∈ Gn are images of a, b ∈ Gn−1 ), and each Gn is obtained from Gn−1 by
adding additional relations between these two generators. Also, the injectivity radius of each surjection
Gn−1 → Gn (i.e., the maximal number r such that the map is injective on all words of length ≤ r) can be
chosen to be arbitrarily large.
Enumerate all the non-trivial words in the free group of countable rank F as v1 , v2 , . . . . Since each Gn
contains a copy of F , there are elements in Gn such that the value of vn on these elements is different
from 1. Writing these elements in terms of the generators a, b, we get
(9)

wn (a, b) = vn (wn1 (a, b), wn2 (a, b), . . .) 6= 1 in Gn

for some (finite number of) words wn , wn1 , wn2 , . . . .
Now, on each step choose the injectivity radius of the surjection Gn → Gn+1 larger than the length of
all words w1 , . . . , wn constructed on the previous steps. Consequently, (9) holds in all groups Gn+1 , Gn+2 , . . . ,
and hence in the limit group G. This implies that vn = 1, for any n, cannot be an identity of G.
In [GG], the absence of nontrivial identities in a Tarski’s monster G of type ∞ is characterized in
terms of an action of the group of outer automorphisms of a free group of rank n on n-tuples of G.
Question. Prove existence of Tarski’s monsters satisfying condition (i) of Theorem 6.1.
Proof of Theorem 6.1. The “only if” part is obvious, so let us prove the “if” part. Let G be Tarski’s monster either of type p which does not satisfy any nontrivial identity except x p = 1 and its consequences, or
of type ∞ which does not satisfy a nontrivial identity. By Corollary 3.2, a group elementarily equivalent
to G contains a subgroup isomorphic to a relatively free subgroup G of rank 2 (which is the free Burnside
group B(2, p) in the case (i), or the free group in the case (ii)). Let x, y be the free generators of G, and
{w1 (x, y) = x, w2 (x, y) = x−1 , w3 (x, y) = y, w4 (x, y) = y−1 , . . . , wkn (x, y)}

ON THE UTILITY OF ROBINSON–AMITSUR ULTRAFILTERS

13

the set of all words of G of length ≤ n. The existence of this “initial piece of G of length n” can be
written as the first-order property:
∃x ∃y :

^

wi (x, y) 6= w j (x, y).

1≤i< j≤kn

Consequently, for each n ∈ N , the same first-order formula holds in G; let xn , yn ∈ G be the corresponding
elements. Obviously, xn , yn do not commute except, possibly, for some small values of n, and, therefore,
generate G. This provides a presentation of G of (relative) girth > n.

Note another interesting consequence of Theorem 6.1.
The growth sequence of a group G is a sequence whose nth term equal to the (minimal) number of
generators of the nth fold direct power of G. See [W] for a brief history of the subject and further references. In particular, in a number of works, including [W], a considerable effort was put into construction
of groups whose growth sequence is constant, each term is equal to 2. Theorem 6.1 provides further
such examples, in view of the following general elementary fact:
Lemma 6.2. If a finitely-generated simple group G has infinite relative girth, then its growth sequence
is constant, each term is equal to the minimal number of generators of G.
Proof. Let n be the minimal number of generators of G. Infinity of the relative girth of G means that
there is an infinite sequence N1 , N2 , . . . of normal subgroups of the free group G in Var(G) of rank n
such that for every i the length of each word in Ni is ≥ i, and G/Ni ≃ G.
Let us prove by induction that for each k ∈ N there is a sequence i1 < i2 < · · · < ik such that
(10)

G/(Ni1 ∩ · · · ∩ Nik ) ≃ G × G × · · · × G

(k times).

For k = 1 we may take i1 = 1. Suppose that for some
k > 1 the isomorphism (10) holds. It is obvious
T
that Ni1 ∩ · · · ∩ Nik 6= {1}. On the other hand, since i>ik Ni = {1}, there is ik+1 > ik such that
Ni1 ∩ · · · ∩ Nik 6⊆ Nik+1 .

Since G is simple, Nik+1 is a maximal normal subgroup in G, and
G = (Ni1 ∩ · · · ∩ Nik )Nik+1 .
Then:
G/(Ni1 ∩ · · · ∩ Nik ∩ Nik+1 ) ≃ G/(Ni1 ∩ · · · ∩ Nik ) × G/Nk+1
≃ G×···×G
It follows from (10) that each finite direct power of G is n-generated.

(k + 1 times).


In fact, nothing in this proof is specific to groups: the corresponding statement can be formulated for
general algebraic systems; in particular, it holds also for algebras.
Lemma 6.2 implies that the growth sequence of Tarski’s monsters satisfying conditions of Theorem
6.1 is constant, each term is equal to 2. Note that in [GG, Theorem 8] it is proved, by different methods,
that each term in the growth sequence of any Tarski’s monster is ≤ 3, and for any Tarski’s monster of
type ∞ it is equal to 2.
7. H OMOMORPHISMS

OF DIRECT PRODUCTS

The following is, in a sense, dual version of Theorem 1.1, with embeddings replaced by surjective
homomorphisms, with, essentially, the same proof†.
Theorem 7.1. Let {Bi }i∈I be a set of algebraic systems from an ideal-determined class. If a finitely
subdirectly irreducible algebraic system A is a surjective homomorphic image of the direct product
∏i∈I Bi , then there is an ultrafilter U on the set I such that A is a surjective homomorphic image of the
ultraproduct ∏U Bi .
†

Added August 22, 2015: the proof of Theorem 7.1 is in error. For a corrected statement, with a different proof, see a
sequel to this article, On the utility of Robinson–Amitsur ultrafilters. II.

14

PASHA ZUSMANOVICH

Proof. Let α : ∏i∈I Bi → A be a given surjective homomorphism. Define
S = {{i ∈ I | f (i) 6= e} | f ∈ ∏ Bi , α ( f ) 6= e}.
i∈I

Let us verify that intersection of any two elements of S contains an element of S . Let S, T ∈ S ,
say, S = {i ∈ I | f (i) 6= e} and T = {i ∈ I | g(i) 6= e} for some f , g ∈ ∏i∈I Bi not vanishing under α .
Since A is finitely subdirectly irreducible, it contains an element a 6= e belonging to the intersection of
ideals generated by {α ( f )} and {α (g)}. Take some element u ∈ ∏i∈I Bi from the preimage α −1 (a).
Let i ∈ I such that f (i) = e. Since α (u) = a = t(b1, . . ., bn , α ( f ), . . ., α ( f )) for some ideal term t and
b1 , . . ., bn ∈ A, we have u = t(h1, . . ., hn , f , . . . , f ) for some h1 ∈ α −1 (b1 ), . . ., hn ∈ α −1 (bn ), and
u(i) = t(h1, . . ., hn , f , . . . , f )(i) = t(h1(i), . . ., hn (i), f (i), . . ., f (i))
= t(h1(i), . . ., hn (i), e, . . ., e) = e.
Coupling this with a similar assertion for g, we get that
S ∩ T ⊃ {i ∈ I | u(i) 6= e} ∈ S .
Thus
 S satisfies
 the finite intersection property and is contained in some ultrafilter U on I. If
f ∈ I ∏i∈I Bi , U , then {i ∈ I | f (i) = e} ∈ U , and, since U is ultrafilter, {i ∈ I | f (i) 6= e} ∈
/ U , and


hence {i ∈ I | f (i) 6= e} ∈
/ S and α ( f ) = e. This shows that I ∏i∈I Bi , U lies in Ker α , and the map

. 

α factors through the ultraproduct ∏i∈I Bi I ∏i∈I Bi , U = ∏U Bi .

In [Zi, §5.3], it is asked whether an ultraproduct of finite groups can be mapped surjectively to a
compact simple Lie group G, in particular, SO(3). According to Theorem 7.1, this is equivalent to the
question whether a direct product of finite groups can be mapped surjectively to G.
Question. What can be said about the class of groups obtained by closure of finite groups with respect
to direct products and homomorphic images? Which simple groups, matrix groups, Lie groups, lie in
this class?
A question when a homomorphic image of a direct product of groups or algebras can be factored
through an ultraproduct, or, more generally, through a product of a finite number of ultraproducts, was
studied also in the recent interesting papers [Be] and [BN]. The emphasis there is, however, different:
one imposes various conditions (such as simplicity, solvability, or nilpotency) not on A but on Bi ’s.
8. F URTHER

SPECULATIONS

Here we indicate some of our initial reasons for looking into all this, of a highly speculative character.
8.1. Lie-algebraic monsters. A problem of existence of Lie-algebraic analogs of Tarski’s monsters
– namely, of infinite-dimensional Lie algebras all whose proper subalgebras are one-dimensional – is,
arguably, one of the most difficult problems in the abstract theory of infinite-dimensional Lie algebras
(see, for example, [G] and [L]). For such hypothetical Lie algebras which do not satisfy a nontrivial
identity, an analog of Theorem 6.1 would hold.
Question. Study the notion of (relative) girth for Lie algebras.
A simpler, but still open problem concerns finite-dimensional Lie algebras all whose proper subalgebras are one-dimensional (called L1 -algebras in the sequel). Over perfect fields, every L1 -algebra has
dimension 2 or 3. In characteristic zero this is an immediate consequence of simple reasonings about
forms of algebras and the classical structure theory, and in the case of the perfect base field of characteristic p > 0 this follows from the powerful results of Premet [Pr1] (for p > 5) and [Pr2] (for p = 5).
See, for example, [G, Proposition 1] or [L, Proposition 3.1] for details.
Over non-perfect fields, we do not have the usual tools of Galois theory to handle forms of algebras, as
in [Pr1], and, to quote [V], “the existence [of an L1 -algebra] of dimension greater than 3 is an interesting

ON THE UTILITY OF ROBINSON–AMITSUR ULTRAFILTERS

15

open problem”. Using reasonings similar to those in §2, it is possible to prove that a prime finitedimensional Lie algebra over an infinite field, all whose abelian subalgebras are 1-dimensional, satisfies
a functional identity of the form
(11)

λ (x, y)u(x, y) + µ (x, y)v(x, y) = 0,

where u and v are certain Lie words in 2 variables, and λ and µ are elements of the base field, depending,
in general, on the elements x and y of the algebra. We conjecture that no simple finite-dimensional Lie
algebra of dimension > 3 can satisfy a nontrivial functional identity of the form (11). If this conjecture
is true, it will prove the absence of L1 -algebras of dimension > 3 over an arbitrary base field.
8.2. The Tits alternative. Another reason was the desire to provide an alternative proof of the celebrated Tits alternative, or for one of its not less celebrated consequences, such as growth dichotomy
for linear group. The Tits alternative claims that a linear group contains either a solvable subgroup of
finite index, or a nonabelian free subgroup. A nice and important, yet admitting a few-lines elementary
proof, result of Platonov [Pl] states that a linear group which has a nontrivial identity, contains a solvable subgroup of finite index. Modulo this result, the proof of the Tits alternative reduces to establishing
that a linear group G which does not satisfy a nontrivial identity, contains a nonabelian free subgroup.
One may naively argue as follows: by Corollary 3.2, a group, elementarily equivalent to G, contains a
nonabelian free subgroup. From this we may infer some first-order properties of G, for example, that
it contains “a piece of a nonabelian free group of arbitrarily large length”, like in the proof of Theorem
6.1. On the other hand, as linearity is a sort of finiteness condition, one may hope that these first-order
properties may help to construct a nonabelian free subgroup in G. If successful, this approach would
provide a proof of the Tits alternative drastically different from all the proofs given so far.
8.3. Jacobson’s problem. An old open problem due to Jacobson asks whether a Lie p-algebra L such
that for every x ∈ L there is n(x) ∈ N satisfying
(12)

n(x)

xp

= x,

is abelian? As one of the first steps, one may wish to prove that such Lie algebras satisfy a nontrivial
identity; or, for example, that there are no simple (or even prime) such algebras. In both of these cases,
by Lemma 2.4 and Corollary 1.2, a free Lie algebra embeds in an algebra elementarily equivalent to
L, over some elementary extension of the base field. The condition (12) is not the first-order property
(unless all n(x) are bounded, in which case the problem is trivial), but one may hope to derive from
it some first-order consequences which will come into contradiction with the existence of a free Lie
subalgebra.
ACKNOWLEDGEMENTS
Thanks are due to Shelly Garion, Misha Gavrilovich, Sergei Ivanov, Ruvim Lipyanski, Alexander
Olshanskii, Mark Sapir, Ivan Shestakov, Irina Sviridova, Jenő Szigeti, and Boris Zilber for helpful and
interesting comments.
A previous version of this manuscript was rejected by Journal of the London Mathematical Society,
but I owe to the anonymous referee of that submission a great deal of improvements and corrections,
both explicit and implicit: in particular, he gave me an idea to use the condition of finite subdirect
irreducibility, and to formulate Theorem 1.1 in the language of universal algebra. The final touches
were done during my visit to IHÉS at February 2012.
This work was partially supported by grants ERMOS7 (Estonian Science Foundation and Marie Curie
Actions) and ETF9038 (Estonian Science Foundation).
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