Von Neumann Regular Cellular Automata
Alonso Castillo-Ramirez and Maximilien Gadouleau

arXiv:1701.02692v3 [] 26 May 2017

May 29, 2017

Abstract
For any group G and any set A, a cellular automaton (CA) is a transformation of the
configuration space AG defined via a finite memory set and a local function. Let CA(G; A)
be the monoid of all CA over AG . In this paper, we investigate a generalisation of the
inverse of a CA from the semigroup-theoretic perspective. An element τ ∈ CA(G; A) is von
Neumann regular (or simply regular ) if there exists σ ∈ CA(G; A) such that τ ◦ σ ◦ τ = τ
and σ ◦ τ ◦ σ = σ, where ◦ is the composition of functions. Such an element σ is called
a generalised inverse of τ . The monoid CA(G; A) itself is regular if all its elements are
regular. We establish that CA(G; A) is regular if and only if |G| = 1 or |A| = 1, and we
characterise all regular elements in CA(G; A) when G and A are both finite. Furthermore,
we study regular linear CA when A = V is a vector space over a field F; in particular, we
show that every regular linear CA is invertible when G is torsion-free elementary amenable
(e.g. when G = Zd , d ∈ N) and V = F, and that every linear CA is regular when V is
finite-dimensional and G is locally finite with char(F) ∤ o(g) for all g ∈ G.
Keywords: Cellular automata, linear cellular automata, monoids, von Neumann regular
elements, generalised inverses.

1

Introduction

Cellular automata (CA), introduced by John von Neumann and Stanislaw Ulam in the 1940s, are
models of computation with important applications to computer science, physics, and theoretical
biology. We follow the modern general setting for CA presented in [5]. For any group G and
any set A, a CA over G and A is a transformation of the configuration space AG defined via
a finite memory set and a local function. Most of the classical literature on CA focus on the
case when G = Zd , for d ≥ 1, and A is a finite set (see [11]), but important results have been
obtained for larger classes of groups (e.g., see [5] and references therein).
Recall that a semigroup is a set equipped with an associative binary operation, and that a
monoid is a semigroup with an identity element. Let CA(G; A) be the set of all CA over G and
A. It turns out that, equipped with the composition of functions, CA(G; A) is a monoid. In
this paper we apply functions on the right; hence, for τ, σ ∈ CA(G; A), the composition τ ◦ σ,
denoted simply by τ σ, means applying first τ and then σ.
In general, τ ∈ CA(G; A) is invertible, or reversible, or a unit, if there exists σ ∈ CA(G; A)
such that τ σ = στ = id. In such case, σ is called the inverse of τ and denoted by σ = τ −1 .
When A is finite, it may be shown that τ ∈ CA(G; A) is invertible if and only if it is a bijective
function (see [5, Theorem 1.10.2]).
We shall consider the notion of regularity which, coincidentally, was introduced by John von
Neumann in the context of rings, and has been widely studied in semigroup theory (recall that
the multiplicative structure of a ring is precisely a semigroup). Intuitively, cellular automaton
τ ∈ CA(G; A) is von Neumann regular if there exists σ ∈ CA(G; A) mapping any configuration
in the image of τ to one of its preimages under τ . Clearly, this generalises the notion of
reversibility.
1

Henceforth, we use the term ‘regular’ to mean ‘von Neumann regular’. Let S be any semigroup. For a, b ∈ S, we say that b is a weak generalised inverse of a if
aba = a.
We say that b is a generalised inverse (often just called an inverse) of a if
aba = a and bab = b.
An element a ∈ S may have none, one, or more (weak) generalised inverses. It is clear that any
generalised inverse of a is also a weak generalised inverse; not so obvious is that, given the set
W (a) of weak generalised inverses of a we may obtain the set V (a) of generalised inverses of a
as follows (see [6, Exercise 1.9.7]):
V (a) = {bab′ : b, b′ ∈ W (a)}.
An element a ∈ S is regular if it has at least one generalised inverse (which is equivalent of having
at least one weak generalised inverse). A semigroup S itself is called regular if all its elements
are regular. Many of the well-known types of semigroups are regular, such as idempotent
semigroups (or bands), full transformation semigroups, and Rees matrix semigroups. Among
various advantages, regular semigroups have a particularly manageable structure which may be
studied using the so-called Green’s relations. For further basic results on regular semigroups
see [6, Section 1.9].
Another generalisation of reversible CA has appeared in the literature before [14, 15] using
the concept of Drazin inverse [8]. However, as Drazin invertible elements are a special kind of
regular elements, our approach turns out to be more general and natural.
In the following sections we study the regular elements in monoids of CA. First, in Section
2 we present some basic results and examples, and we establish that, except for the trivial
cases |G| = 1 and |A| = 1, the monoid CA(G; A) is not regular. In Section 3, we study the
regular elements of CA(G; A) when G and A are both finite; in particular, we characterise them
and describe a regular submonoid. In Section 4, we study the regular elements of the monoid
LCA(G; V ) of linear CA, when V is a vector space over a field F. Specifically, using results
on group rings, we show that, when G is torsion-free elementary amenable (e.g., G = Zd ),
τ ∈ LCA(G; F) is regular if and only if it is invertible, and that, for finite-dimensional V ,
LCA(G; V ) itself is regular if and only if G is locally finite and char(F) ∤ |hgi|, for all g ∈ G.
Finally, for the particular case when G ∼
= Zn is a cyclic group, V := F is a finite field, and
char(F) | n, we count the total number of regular elements in LCA(Zn ; F).

2

Regular cellular automata

For any set X, let Tran(X), Sym(X), and Sing(X), be the sets of all functions, all bijective
functions, and all non-bijective (or singular) functions of the form τ : X → X, respectively.
Equipped with the composition of functions, Tran(X) is known as the full transformation
monoid on X, Sym(X) is the symmetric group on X, and Sing(X) is the singular transformation semigroup on X. When X is a finite set of size α, we simply write Tranα , Symα , and
Singα , in each case.
We shall review the broad definition of CA that appears in [5, Sec. 1.4]. Let G be a group
and A a set. Denote by AG the configuration space, i.e. the set of all functions of the form
x : G → A. For each g ∈ G, denote by Rg : G → G the right multiplication function, i.e.
(h)Rg := hg for any h ∈ G. We emphasise that we apply functions on the right, while [5]
applies functions on the left.

2

Definition 1. Let G be a group and A a set. A cellular automaton over G and A is a transformation τ : AG → AG satisfying the following: there is a finite subset S ⊆ G, called a memory
set of τ , and a local function µ : AS → A such that
(g)(x)τ = ((Rg ◦ x)|S )µ, ∀x ∈ AG , g ∈ G,
where (Rg ◦ x)|S is the restriction to S of (Rg ◦ x) : G → A .
The group G acts on the configuration space AG as follows: for each g ∈ G and x ∈ AG , the
configuration x · g ∈ AG is defined by
(h)x · g := (hg −1 )x,

∀h ∈ G.

A transformation τ : AG → AG is G-equivariant if, for all x ∈ AG , g ∈ G,
(x · g)τ = ((x)τ ) · g.
Any cellular automaton is G-equivariant, but the converse is not true in general. A generalisation
of Curtis-Hedlund Theorem (see [5, Theorem 1.8.1]) establishes that, when A is finite, τ : AG →
AG is a CA if and only if τ is G-equivariant and continuous in the prodiscrete topology of AG ;
in particular, when G and A are both finite, G-equivariance completely characterises CA over
G and A.
A configuration x ∈ AG is called constant if (g)x = k, for a fixed k ∈ A, for all g ∈ G. In
such case, we denote x by k ∈ AG .
Remark 1. It follows by G-equivariance that any τ ∈ CA(G; A) maps constant configurations
to constant configurations.
Recall from Section 1 that τ ∈ CA(G; A) is invertible if there exists σ ∈ CA(G; A) such
that τ σ = στ = id, and that τ ∈ CA(G; A) is regular if there exists σ ∈ CA(G; A) such that
τ στ = τ . We now present some examples of CA that are regular but not invertible.
Example 1. Let G be any nontrivial group and A any set with at least two elements. Let
σ ∈ CA(G; A) be a CA with memory set {s} ⊆ G and local function µ : A → A that is nonbijective. Clearly, σ is not invertible. As Sing(A) is a regular semigroup (see [10, Theorem II]),
there exists µ′ : A → A such that µµ′ µ = µ. If σ ′ is the CA with memory set {s−1 } and local
function µ′ , then σσ ′ σ = σ. Hence σ is regular.
Example 2. Suppose that A = {0, 1, . . . , q − 1}, with q ≥ 2. Consider τ1 , τ2 ∈ CA(Z; A) with
memory set S := {−1, 0, 1} and local functions
(x)µ1 = min{(−1)x, (0)x, (1)x} and (x)µ2 = max{(−1)x, (0)x, (1)x},
respectively, for all x ∈ AS . Clearly, τ1 and τ2 are not invertible, but we show that they are
generalised inverses of each other, i.e. τ1 τ2 τ1 = τ1 and τ2 τ1 τ2 = τ2 , so they are both regular.
We prove only the first of the previous identities, as the second one is symmetrical. Let x ∈ AZ ,
y := (x)τ1 , z := (y)τ2 , and a := (z)τ1 . We want to show that y = a. For all i ∈ Z and
ǫ ∈ {−1, 0, 1}, we have
(i + ǫ)y = min{(i + ǫ − 1)x, (i + ǫ)x, (i + ǫ + 1)x} ≤ (i)x.
Hence,
(i)z = max{(i − 1)y, (i)y, (i + 1)y} ≤ (i)x.
Similarly (i − 1)z ≤ (i − 1)x and (i + 1)z ≤ (i + 1)x, so
(i)a = min{(i − 1)z, (i)z, (i + 1)z} ≤ (i)y = min{(i − 1)x, (i)x, (i + 1)x}.
Conversely, we have (i − 1)z, (i)z, (i + 1)z ≥ (i)y, so (i)a ≥ (i)y. In particular, when q = 2, τ1
and τ2 are the elementary CA known as Rules 128 and 254, respectively.
3

The following lemma gives an equivalent definition of regular CA. Note that this result still
holds if we replace CA(G; A) with any monoid of transformations.
Lemma 1. Let G be a group and A a set. Then, τ ∈ CA(G; A) is regular if and only if there
exists σ ∈ CA(G; A) such that for every y ∈ (AG )τ there is ŷ ∈ AG with (ŷ)τ = y and (y)σ = ŷ.
Proof. If τ ∈ CA(G; A) is regular, there exists σ ∈ CA(G; A) such that τ στ = τ . Let x ∈ AG
be such that (x)τ = y (which exists because y ∈ (AG )τ ) and define ŷ := (y)σ. Now,
(ŷ)τ = (y)στ = (x)τ στ = (x)τ = y.
Conversely, assume there exists σ ∈ CA(G; A) satisfying the statement of the lemma. Then, for
any x ∈ AG with y := (x)τ we have
(x)τ στ = (y)στ = (ŷ)τ = y = (x)τ.
Therefore, τ is regular.
Corollary 1. Let G be a nontrivial group and A a set with at least two elements. Let τ ∈
CA(G; A), and suppose there is a constant configuration k ∈ (AG )τ such that there is no constant
configuration of AG mapped to k under τ . Then τ is not regular.
Proof. The result follows by Remark 1 and Lemma 1.
In the following examples we see how Corollary 1 may be used to show that some well-known
CA are not regular.
Example 3. Let φ ∈ CA(Z; {0, 1}) be the Rule 110 elementary CA, and consider the constant
configuration 1. Define x := . . . 10101010 · · · ∈ {0, 1}Z , and note that (x)φ = 1. Since (1)φ = 0
and (0)φ = 0, Corollary 1 implies that φ is not regular.
Example 4. Let τ ∈ CA(Z2 ; {0, 1}) be Conway’s Game of Life, and consider the constant
configuration 1 (all cells alive). By [5, Exercise 1.7.], 1 is in the image of τ ; since (1)τ = 0 (all
cells die from overpopulation) and (0)τ = 0, Corollary 1 implies that τ is not regular.
The following theorem applies to CA over arbitrary groups and sets, and it shows that,
except for the trivial cases, CA(G; A) always contains non-regular elements.
Theorem 1. Let G be a group and A a set. The semigroup CA(G; A) is regular if and only if
|G| = 1 or |A| = 1.
Proof. If |G| = 1 or |A| = 1, then CA(G; A) = Tran(A) or CA(G; A) is the trivial semigroup
with one element, respectively. In both cases, CA(G; A) is regular (see [6, Exercise 1.9.1]).
Assume that |G| ≥ 2 and |A| ≥ 2. Suppose that {0, 1} ⊆ A. Let S := {e, g, g −1 } ⊆ G, where
e is the identity of G and e 6= g ∈ G (we do not require g 6= g −1 ). For i = 1, 2, let τi ∈ CA(G; A)
be the cellular automaton defined by the local function µi : AS → A, where, for any x ∈ AS ,
(
(e)x if (e)x = (g)x = (g−1 )x,
(x)µ1 :=
0
otherwise;
(
1
if (e)x = (g)x = (g−1 )x = 0,
(x)µ2 :=
(e)x otherwise.
We shall show that τ := τ2 τ1 ∈ CA(G; A) is not regular.
Consider the constant configurations 0, 1 ∈ AG . Let z ∈ AG be defined by
(
m mod (2) if h = gm , m ∈ N minimal,
(h)z :=
0
otherwise.
4

τ2

z
τ1 , τ2

τ1
τ1

0

k
τ2
τ1 , τ2

1

Figure 1: Images of τ1 and τ2 .
Figure 1 illustrates the images z, 0, 1, and k 6= 0, 1 (in case it exists) under τ1 and τ2 .
Clearly,
(0)τ = (0)τ2 τ1 = (1)τ1 = 1.
In fact,
(k)τ =

(

1 if k = 0,
k otherwise.

Furthermore,
(z)τ = (z)τ2 τ1 = (z)τ1 = 0.
Hence, 0 is a constant configuration in the image of τ but with no preimage among the constant
configurations. By Corollary 1, τ is not regular.
Now that we know that CA(G; A) always contains both regular and non-regular elements
(when |G| ≥ 2 and |A| ≥ 2), an interesting problem is to find a criterion that describes all
regular CA. In the following sections, we solve this problem by adding some extra assumptions,
such as finiteness and linearity.

3

Regular finite cellular automata

In this section we characterise the regular elements in the monoid CA(G; A) when G and A
are both finite (Theorem 3). In order to achieve this, we summarise some of the notation and
results obtained in [2, 3, 4].
Definition 2. The following definitions apply for an arbitrary group G and an arbitrary set A:
1. For any x ∈ AG , the G-orbit of x in AG is xG := {x · g : g ∈ G}.
2. For any x ∈ AG , the stabiliser of x in G is Gx := {g ∈ G : x · g = x}.
3. A subshift of AG is a subset X ⊆ AG that is G-invariant, i.e. for all x ∈ X, g ∈ G, we
have x · g ∈ X, and closed in the prodiscrete topology of AG .
4. The group of invertible cellular automata over G and A is
ICA(G; A) := {τ ∈ CA(G; A) : ∃φ ∈ CA(G; A) such that τ φ = φτ = id}.
In the case when G and A are both finite, every subset of AG is closed in the prodiscrete
topology, so the subshifts of AG are simply unions of G-orbits. Moreover, as every map τ :

5

AG → AG is continuous in this case, CA(G; A) consists of all the G-equivariant maps of AG .
Theorem 2 is easily deduced from Lemmas 3, 9 and 10 in [4].
If M is a group, or a monoid, write K ≤ M if K is a subgroup, or a submonoid, of M ,
respectively.
Theorem 2. Let G be a finite group of size n ≥ 2 and A a finite set of size q ≥ 2. Let x, y ∈ AG .
(i) Let τ ∈ CA(G; A). If (x)τ ∈ (xG), then τ |xG ∈ Sym(xG).
(ii) There exists τ ∈ ICA(G; A) such that (x)τ = y if and only if Gx = Gy .
(iii) There exists τ ∈ CA(G; A) such that (x)τ = y if and only if Gx ≤ Gy .
Theorem 3. Let G be a finite group and A a finite set of size q ≥ 2. Let τ ∈ CA(G; A). Then,
τ is regular if and only if for every y ∈ (AG )τ there is x ∈ AG such that (x)τ = y and Gx = Gy .
Proof. First, suppose that τ is regular. By Lemma 1, there exists φ ∈ CA(G; A) such that for
every y ∈ (AG )τ there is ŷ ∈ AG with (ŷ)τ = y and (y)φ = ŷ. Take x := ŷ. By Theorem 2,
Gx ≤ Gy and Gy ≤ Gx . Therefore, Gx = Gy .
Conversely, suppose that for every y ∈ (AG )τ there is x ∈ AG such that (x)τ = y and
Gx = Gy . Choose pairwise distinct G-orbits y1 G, . . . , yℓ G such that
(AG )τ =

ℓ
[

yi G.

i=1

For each i, fix yi′ ∈ AG such that (yi′ )τ = yi and Gyi = Gyi′ . We define φ : AG → AG as follows:
for any z ∈ AG ,
(
z
if z 6∈ (AG )τ,
(z)φ :=
yi′ · g if z = yi · g ∈ yi G.
The map φ is well-defined because
yi · g = yi · h ⇐⇒ hg −1 ∈ Gyi = Gyi′ ⇐⇒ yi′ · g = yi′ · h.
Clearly, φ is G-equivariant, so φ ∈ CA(G; A). Now, for any x ∈ AG with (x)τ = yi · g,
(x)τ φτ = (yi · g)φτ = (yi′ · g)τ = (yi′ )τ · g = yi · g = (x)τ.
This proves that τ φτ = τ , so τ is regular.
Our goal now is to find a regular submonoid of CA(G; A) and describe its structure (see
Theorem 4). In order to achieve this, we need some further terminology and basic results.
Say that two subgroups H1 and H2 of G are conjugate in G if there exists g ∈ G such that
−1
g H1 g = H2 . This defines an equivalence relation on the subgroups of G. Denote by [H] the
conjugacy class of H ≤ G. Define the box in AG corresponding to [H], where H ≤ G, by
B[H] (G; A) := {x ∈ AG : [Gx ] = [H]}.
As any subgroup of G is the stabiliser of some configuration in AG , the set {B[H] (G; A) : H ≤ G}
is a partition of AG . Note that B[H] (G; A) is a subshift of AG (because G(x·g) = g −1 Gx g) and, by
the Orbit-Stabiliser Theorem, all the G-orbits contained in B[H] (G; A) have equal sizes. When
G and A are clear from the context, we write simply B[H] instead of B[H] (G; A).
Example 5. For any finite group G and finite set A of size q, we have
B[G] = {k ∈ AG : k is constant}.
6

For any subshift C ⊆ AG , define
CA(C) := {τ ∈ Tran(C) : τ is G-equivariant}.
In particular, CA(AG ) = CA(G; A). Clearly,
CA(C) = {τ |C : τ ∈ CA(G; A), τ (C) ⊆ C}.
A submonoid R ≤ M is called maximal regular if there is no regular monoid K such that
R < K < M.
Theorem 4. Let G be a finite group and A a finite set of size q ≥ 2. Let

R := σ ∈ CA(G; A) : Gx = G(x)σ for all x ∈ AG .

(i) ICA(G; A) ≤ R.

(ii) R is a regular monoid.
Q
(iii) R ∼
= H≤G CA(B[H] ).

(iv) R is not a maximal regular submonoid of CA(G; A).

Proof. Part (i) and (iii) are trivial while part (ii) follows by Theorem 3.
For part (iv), let x, y ∈ AG be such that Gx < Gy , so x and y are in different boxes.
Define τ ∈ CA(G; A) such that (x)τ = y, (B[Gy ] )τ = yG, and τ fixes any other configuration in
AG \ (B[Gy ] ∪ {xG}). It is clear by Theorem 3 that τ is regular. We will show that K := hR, τ i
is a regular submonoid of CA(G; A). Let σ ∈ K and z ∈ (AG )σ. If σ ∈ R, then it is obviously
regular, so assume that σ = ρ1 τ ρ2 with ρ1 ∈ K and ρ2 ∈ R. If z ∈ AG \ (B[Gy ] ), it is clear that
z has a preimage in its own box; otherwise (B[Gy ] )σ = (yG)ρ2 = zG and z has a preimage in
B[Gy ] . Hence σ is regular and so is K.

4

Regular linear cellular automata

Let V a vector space over a field F. For any group G, the configuration space V G is also a
vector space over F equipped with the pointwise addition and scalar multiplication. Denote by
EndF (V G ) the set of all F-linear transformations of the form τ : V G → V G . Define
LCA(G; V ) := CA(G; V ) ∩ EndF (V G ).
Note that LCA(G; V ) is not only a monoid, but also an F-algebra (i.e. a vector space over F
equipped with a bilinear binary product), because, again, we may equip LCA(G; V ) with the
pointwise addition and scalar multiplication. In particular, LCA(G; V ) is also a ring.
As in the case of semigroups, von Neumann regular rings have been widely studied and
many important results have been obtained. In this chapter, we study the regular elements of
LCA(G; V ) under some natural assumptions on the group G.
First, we introduce some preliminary results and notation. The group ring R[G] is the set
of all functions f : G → R with finite support (i.e. the set {g ∈ G : (g)f 6= 0} P
is finite).
Equivalently, the group ring R[G] may be defined as the set of all formal finite sums g∈G ag g
with ag ∈ R. The multiplication in R[G] is defined naturally using the multiplications of G and
R:
X
X
X
ag g
ah h =
ag ah gh.
g∈G

h∈G

g,h∈G

If we let R := EndF (V ), it turns out that EndF (V )[G] is isomorphic to LCA(G; V ) as F-algebras
(see [5, Theorem 8.5.2]).
7

Define the order of g ∈ G by o(g) := |hgi| (i.e. the size of the subgroup generated by g).
The group G is torsion-free if the identity is the only element of finite order; for instance, the
groups Zd , for d ∈ N, are torsion-free groups. The group G is elementary amenable if it may be
obtained from finte groups or abelian groups by a sequence of group extensions or direct unions.
In the following theorem we characterise the regular linear cellular automata over fields and
torsion-free elementary amenable groups (such as Zd , d ∈ N).
Theorem 5. Let G be a torsion-free elementary amenable group and let V = F be any field. A
non-zero element τ ∈ LCA(G; F) is regular if and only if it is invertible.
Proof. It is clear that any invertible element is regular. Let τ ∈ LCA(G; F) be non-zero regular.
In this case, EndF (F) ∼
= F, so LCA(G; F) ∼
= F[G]. By definition, there exists σ ∈ LCA(G; V )
such that τ στ = τ . As LCA(G; V ) is a ring, the previous is equivalent to
τ (στ − 1) = 0,
where 1 = 1e and 0 = 0e are the identity and zero endomorphisms, respectively. Since τ 6= 0,
either στ − 1 = 0, in which case τ is invertible, or στ − 1 is a zero-divisor. However, it was
established in [12, Theorem 1.4] that F[G] has no zero-divisors whenever G is a torsion-free
elementary amenable group. Hence, τ is invertible.
The argument of the previous result works as long as the group ring F[G] has no zero-divisor.
This is connected with the well-known Kaplansky’s conjecture which states that F[G] has no
zero-divisors when G is a torsion-free group.
The characteristic of a field F, denoted by char(F), is the smallest k ∈ N such that
1| + 1 +
{z· · · + 1} = 0,
k times

where 1 is the multiplicative identity of F. If no such k exists we say that F has characteristic
0.
A group G is locally finite if every finitely generated subgroup of G is finite; in particular,
the order of every element of G is finite. Examples of such groups are finite groups and infinite
direct sums of finite groups.
Theorem 6. Let G be a group and let V be a finite-dimensional vector space over F. Then,
LCA(G; V ) is regular if and only if G is locally finite and char(F) ∤ o(g), for all g ∈ G.
Proof. By [7, Theorem 3] (see also [1, 13]), we have that a group ring R[G] is regular if and
only if R is regular, G is locally finite and o(g) is a unit in R for all g ∈ G. In the case of
LCA(G; V ) ∼
= EndF (V )[G], since dim(V ) := n < ∞, the ring R := EndF (V ) ∼
= Mn×n (F) is
regular (see [9, Theorem 1.7]. The condition that o(g), seen as the matrix o(g)In , is a unit in
Mn×n (F) is satisfied if and only if o(g) is nonzero in F, which is equivalent to char(F) ∤ o(g), for
all g ∈ G.
Corollary 2. Let G be a group and let V be a finite-dimensional vector space over a field F of
characteristic 0. Then, LCA(G; V ) is regular if and only if G is locally finite.
Henceforth, we focus on the regular elements of LCA(G; V ) when V is a one-dimensional
vector space (i.e. V is just the field F). In this case, EndF (F) ∼
= F, so LCA(G; F) and F[G] are
isomorphic as F-algebras.
A non-zero element a of a ring R is called nilpotent if there exists n > 0 such that an = 0.
The following basic result will be quite useful in the rest of this section.
Lemma 2. Let R be a commutative ring. If a ∈ R is nilpotent, then a is not a regular element.
8

Proof. Let R be a commutative ring and a ∈ R a nilpotent element. Let n > 0 be the smallest
integer such that an = 0. Suppose a is a regular element, so there is x ∈ R such that axa = a.
By commutativity, we have a2 x = a. Multiplying both sides of this equation by an−2 we obtain
0 = an x = an−1 , which contradicts the minimality of n.
Example 6. Suppose that G is a finite abelian group and let F be a field such that char(F) |
|G|. P
By Theorem 6, LCA(G; F) must have elements that are not regular. For example, let
s := g∈G g ∈ F[G]. As sg = s, for all g ∈ G, and char(F) | |G|, we have s2 = |G|s = 0.
Clearly, F[G] is commutative because G is abelian, so, by Lemma 2, s is not a regular element.
We finish this section with the special case when G is the cyclic group Zn and F is a finite
field with char(F) | n. By Theorem 6, not all the elements of LCA(Zn ; F) are regular, so
how many of them are there? In order to count them we need a few technical results about
commutative rings.
An ideal I of a commutative ring R is a subring such that rb ∈ I for all r ∈ R, b ∈ I. For
any a ∈ R, the principal ideal generated by a is the ideal hai := {ra : r ∈ R}. A ring is called
local if it has a unique maximal ideal.
Denote by F[x] the ring of polynomials with coefficients in F. When G ∼
= Zn , we have the
following isomorphisms as F-algebras:
LCA(Zn ; F) ∼
= F[Zn ] ∼
= F[x]/hxn − 1i,
where hxn − 1i is a principal ideal in F[x].
Theorem 7. Let n ≥ 2 be an integer, and let F be a finite field of size q such that char(F) | n.
Consider the following factorization of xn − 1 into irreducible elements of F[x]:
xn − 1 = p1 (x)m1 p2 (x)m2 . . . pr (x)mr .
For each i = 1, . . . r, let di := deg(pi (x)). Then, the number of regular elements in LCA(Zn ; F)
is exactly
r 

Y
(q di − 1)q di (mi −1) + 1 .
i=1

Proof. Recall that

LCA(Zn ; F) ∼
= F[x]/hxn − 1i.

By the Chinese Remainder Theorem,
F[x]/hxn − 1i ∼
= F[x]/hp1 (x)m1 i × F[x]/hp2 (x)m2 i × · · · × F[x]/hpr (x)mr i.
An element a = (a1 , . . . , ar ) in the right-hand side of the above isomorphism is a regular element
if and only if ai is a regular element in F[x]/hpi (x)mi i for all i = 1, . . . , r.
Fix m := mi , p(x) = pi (x), and d := di . Consider the principal ideals A := hp(x)i and
B := hp(x)m i in F[x]. Then, F[x]/B is a local ring with unique maximal ideal A/B, and
each of its nonzero elements is either nilpotent or a unit (i.e. invertible): in particular, the
set of units of F[x]/B is precisely (F[x]/B) − (A/B). By the Third Isomorphism Theorem,
(F[x]/B)/(A/B) ∼
= (F[x]/A), so
|A/B| =

q dm
|F[x]/B|
= d = q d(m−1) .
|F[x]/A|
q

Thus, the number of units in F[x]/B is
|(F[x]/B) − (A/B)| = q dm − q d(m−1) = (q d − 1)q d(m−1) .
As nilpotent elements are not regular by Lemma 2, every regular element of F[x]/hpi (x)mi i is
zero or a unit. Thus, the number of regular elements in F[x]/hpi (x)mi i is (q di −1)q di (mi −1) +1.
9

Acknowledgments.
We thank the referees of this paper for their insightful suggestions and corrections. In particular,
we thank the first referee for suggesting the references [1, 7, 13], which greatly improved the
results of Section 4.

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A. Castillo-Ramirez (Corresponding author), Universidad de Guadalajara, CUCEI, Departamento de
Matemáticas, Guadalajara, México.
Email: alonso.castillor@academicos.udg.mx
M. Gadouleau, School of Engineering and Computing Sciences, Durham University, South Road,
Durham DH1 3LE, U.K.
Email: m.r.gadouleau@durham.ac.uk

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