arXiv:1612.02476v1 [] 7 Dec 2016

I-PRIMARY SUBMODULES
ISMAEL AKRAY and HALGURD S. HUSSEIN
Abstract
In this paper, we give a generalization for weakly primary submodules called I-primary submodule
and we study some properties of it. We give some characterizations of I-primary submodules. Also we
establish the situation of I-primary submodules in module localizations and decomposable modules.
2010 Mathematics subject classification: primary 13A15; secondary 13C99; 13F05.
Keywords and phrases: Primary ideal, primary submodule, weakly primary submodule.

1. Introduction
Throughout this paper R will be a commutative ring with nonzero identity and I a
fixed ideal of R and M a unitary left R-module. The concept of weakly prime ideals
was introduced by Anderson and Smith (2003), where a proper ideal P is called weakly
prime if, for a, b ∈ R with 0 , ab ∈ P, either a ∈ P or b ∈ P, [5]. The radical of an
ideal I of R is defined to be the set of all a ∈ R for which an ∈ I for some positive
integer n. Primary ideals have an important role in commutative ring theory. A proper
ideal Q of R is said
√ to be primary provided that for a, b ∈ R, ab ∈ Q implies that either
a ∈ Q or b ∈ Q. We can generalize the concept of primary ideals by restricting
the set where ab lies. A proper √ideal Q of R is weakly primary if for a, b ∈ R with
0 , ab ∈ Q, either a ∈ Q or b ∈ Q. Weakly primary ideals were first introduced and
studied by Ebrahimi Atani and Farzalipour in 2005, [9].
An R-module M is called a multiplication module if every submodule N of M has
the form IM for some ideal I of R, see [3]. Note that, since I ⊆ (N :R M) then
N = IM ⊆ (N :R M)M ⊆ N. So that N = (N :R M)M. Let N and K be submodules
of a multiplication R-module M with N = I1 M and K = I2 M for some ideals I1 and I2
of R. The product of N and K denoted by NK is defined by NK = I1 I2 M. Then by [4,
Theorem 3.4], the product of N and K is independent of presentations of N and K. An
R-module M is called faithful if it has zero annihilator. Let N be a proper submodule
of a nonzero R-module M. Then the M-radical of N, denoted by M − rad(N), is
defined to be the intersection of all prime submodules of M containing N. If M has
no prime submodule containing N, then we say M − rad(N) = M. It is shown in [10,
Theorem 2.12] √
that if N is a proper submodule of a multiplication R-module M, then
M − rad(N) = (N :R M)M.

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I. Akray and H. S. Hussein

The class of prime submodules of modules was introduced and studied in 1992 as
a generalization of the class of prime ideals of rings. Then, many generalizations of
prime submodules were studied such as primary, classical prime, weakly prime and
classical primary submodules, see [3, 7, 8, 11] and [6]. The authors in [1] and [2]
introduced the notions I-prime ideals and I-prime submodules. A proper ideal of R is
called I-prime ideal of R if rs ∈ P − IP for all r, s ∈ R implies that either r ∈ P or
s ∈ P. A proper submodule P of M is called I-prime submodule of M if rm ∈ P − IP
for all r ∈ R and m ∈ M implies that either m ∈ P or r ∈ (P : M). In this article, we
define and study I-primary submodules which are generalizations of weakly primary
(and weakly prime) submodules. We generalize some basic properties of weakly prime
and weakly primary to I-primary submodules. We give some characterizations of Iprimary submodules.

2. Main results
Let I be an ideal of R and M an R-module. A proper submodule P of M is called
an I -primary submodule
of M if rm ∈ P − IP for all r ∈ R and m ∈ M implies that
√
either m ∈ P or r ∈ (P : M). In view of the above definition, in all what follows, I is
an ideal of R. Now we begin with the following result.
Theorem 2.1. Let M be an R-module. Let P be an I-primary submodule of M. If
(P : M)P * IP, then P is a primary submodule of M.
Proof. Suppose that r ∈ R and m ∈ M such that rm ∈ P. If rm < IP, since P is
I-primary submodule of M, then rn ∈ (P : M) for some positive integer n or m ∈ P.
Now suppose that rm ∈ IP. Let rP ⊆ IP. Because if rP * IP, then there exists p ∈ P
such that r p < IP, then r(m + p) ∈ P − IP. Therefore rn ∈ (P : M) for some positive
integer n or m + p ∈ P. Hence rn ∈ (P : M) for some positive integer n or m ∈ p. Now
suppose that (P : M)m ⊆ IP. Because if (P : M)m * IP, then there√exists t ∈ (P : M)
such that tm √
< IP and so (r + t)m ∈ P − IP. Then we have r + t ∈ (P : M) or m ∈ P
. Hence r ∈ (P : M) or m ∈ P. Since (P : M)P * IP, so there exist b ∈ (P √: M) and
p ∈ P such that bp < IP. √
Then (r + b)(m + p) ∈ P − IP. Therefore r + b ∈ (P : M)
or m + p ∈ P. Hence r ∈ (P : M) or m ∈ P. Thus P is primary submodule in M. 
Corollary 2.2. Let P be a 0-primary of M such that (P : M)P , 0. Then P is a
primary submodule of M.
Corollary 2.3. Let P be an I-primary submodule of M such that IP ⊆ (P : M)2 P and
that P is not primary submodule. Then IP = (P : M)2 P.
In what follows we give some charactrizations for I-primary submodules.
Theorem 2.4. Let P be a proper submodule of M, then the following are equivalent:
(1) P is an I-primary
submodule of M.
√
(2)For r ∈ R − √(P : M), (P : r) = P ∪ (IP : r).
(3) For r ∈ R − (P : M), (P : r) = P or (P : r) = (IP : r).

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3

Proof. (1)⇒ (2) Suppose that P is an I-primary submodule of M such that r <
√
(P : M). Let a ∈ (P : r). So ra ∈ P. If ra < IP, then a ∈ P. Because P is an
I-primary submodule in M. If ra ∈ IP, then a ∈ (IP : r). So (P : r) ⊆ P ∪ (IP : r).
Now since IP ⊆ P, the other inclusion is hold.
(2)⇒ (3) It is clear because (P : r) is a submodule of M.
√
(3)⇒ (1) Let r ∈ R and x ∈ M such that rx ∈ P − IP. If r < (P : M), then by
assumption, either (P : r) = P or (P : r) = (IP : r). Since rx < IP then x < (IP : r),
and since rx ∈ P, then x ∈ (P : r). So (P : r) = P. Therefore x ∈ P. Thus P is an
I-primary submodule of M.

The quotient and localization of primary submodules are again primary submodules. But in case of I-primary submodules, we give a condition under which the localization becomes true as we see in the following theorem.
Theorem 2.5. Let M be an R-module. Let P be an I-primary submodule of M. Then,
(i) If N ⊆ P is a submodule of M, then NP is an I-primary submodule of M
N.
(ii) Suppose that S is a multiplicatively closed subset of R such that S −1 P , S −1 M and
S −1 (IP) ⊆ (S −1 I)(S −1 P). Then S −1 P is an (S −1 I)-primary submodule of S −1 M.
Proof. (i) Let r ∈ R and m ∈ M such that r(m + N) = rm + N ∈ NP − I NP . Then
n
rm + N ∈ P−IP
N . So rm ∈ P − IP. As P is I-primary submodule, then either r ∈ (P : M)
M
P
n
for some positive integer n or m ∈ P. Therefore r ∈ ( N : N ) for some positive integer
n or m + N ∈ NP . Hence NP is I-primary submodule of M
N.
−1
−1
−1
(ii) For all rs ∈ S −1 R and mt ∈ S −1 M, let rs . mt = rm
st ∈ S P − (S I)(S P) ⊆
x
rm
−1
−1
−1
S P − S (IP) = S (P − IP). Then st = u for x ∈ P − IP and u ∈ S . So for some
q ∈ S , qurm = qstx ∈ P − IP. As P is I-primary
submodule , (qru)n ∈ (P : M) for
(qur)n
r n
some positive integer n or m ∈ P. So (qus)n = ( s ) ∈ S −1 (P : M) = (S −1 P :S −1 R S −1 M)

or mt ∈ S −1 P. Hence S −1 P is (S −1 I)-primary submodule of S −1 M.
Proposition 2.6. Let M be an R-module and P be a submodule of M.
(i) If I1 ⊆ I2 . Then P is I1 -primary implies P is I2 -primary.
(ii) If P is I-prime then P is I-primary.
Proof. (1) Suppose that P is I1 -primary.Let r ∈ R, m ∈ M with rm ∈ P − I2√P. Since
I1 ⊆ I2 , P − I2 P ⊆ P − I1 P. Then rm ∈ P − I1 P. But P is I1 -primary. So r ∈ (P : M)
or m ∈ P. Thus P is I2 -primary.
(2) The proof is trivial.

√
An ideal I is called radical if I = I. In the following we give a condition under
which I-prime and I-primary be equivalent.
Proposition 2.7. Let P be a proper submodule of an R-module M such that (P : M) is
radical ideal. Then P is I-primary if and only if P is I-prime.
Proof. The proof comes from the definitions.



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I. Akray and H. S. Hussein

Theorem 2.8. Let M1 and M2 be modules over R1 and R2 respectively with M =
M1 × M2 . Suppose that P1 is an I1 -primary submodule of M1 such that IP1 × M2 ⊆
I(P1 × M2 ). Then P1 × M2 is an I-primary submodule of M1 × M2 .
Proof. Let (r1 , r2 ) ∈ R, and (x1 , x2 ) ∈ M with (r1 , r2)(x1 , x2 ) = (r1 x1 , r2 x2 ) ∈
P1 × M2 −I(P1 × M2 ), and P1 × M2 −I(P1 × M2 ) ⊆ P1 × M2 −IP1 × M2 = (P1 −IP1 )× M2 .
We have r1 x1 ∈ P1 − IP1 but P1 is I-primary submodule of M1 . Then r1n ∈ (P1 :R1 M1 )
for some positive integer n or x1 ∈ P1 . So (r1 , r2 )n = (r1n , r2n ) ∈ (P1 :R1 M1 ) × R2 =
(P1 × M2 :R1 ×R2 M1 × M2 ) for some positive integer n or (x1 , x2 ) ∈ P1 × M2 . Hence
P1 × M2 is an I-primary submodule of M1 × M2 .

We illustrate the formation of the I-primary submodules in decomposition modules. In other manner the formation of I-primary submodules in decomposable module
have one of the five types stated in the following theorem.
Theorem 2.9. Let R = R1 × R2 . Let M1 and M2 be R1 and R2 -modules respectively
with M = M1 × M2 . Let I1 and I2 be ideals of R1 and R2 respectively with I = I1 × I2 .
Then all the following types are I-primary submodule of M1 × M2 .
1.
2.
3.
4.
5.

P1 × P2 where Pi is a proper submodule of Mi with Ii Pi = Pi for i = 1, 2.
P1 × M2 where P1 is a primary submodule of M1 .
P1 × M2 where P1 is an I1 -primary submodule of M1 and I2 M2 = M2 .
M1 × P2 where P2 is a primary submodule of M2 .
M1 × P2 where P2 is an I2 -primary submodule of M2 and I1 M1 = M1 .

Proof. 1.
Since I1 P1 = P1 and I2 P2 = P2 . Then I1 P1 × I2 P2 = I 1 × I2 (P1 × P2 ) =
I(P1 × P2 ) = P1 × P2 . So P1 × P2 = I(P1 × P2 ) = φ. Thus there is nothing to
prove.
2.
Let P1 be a primary submodule of M1 . Then P1 × M2 is a primary submodule of
M1 × M2 . Thus P1 × M2 is I-primary submodule of M.
3.
Suppose that P1 is an I1 -primary submodule of M1 and I2 M2 = M2 . Let
(r1 , r2 ) ∈ R and (m1 , m2 ) ∈ M such that (r1 , r2 )(m1 , m2) = (r1 m1 , r2 m2 ) ∈
P1 × M2 − I(P1 × M2 ) = P1 × M2 − (I1 × I2 )(P1 × M2 ) = P1 × M2 − (I1 P1 × I2 M2 ) =
P1 × M2 − (I1 P1 × M2 ) = (P1 − I1 P1 ) × M2 . Then r1 m1 ∈ P1 − I1 P1 and P1 is
I1 -primary submodule of M1 , so r1n ∈ (P1 :R1 M1 ) for some positive integer n ∈ N
or m1 ∈ P1 . Therefore (r1 , r2 )n = (r1n , r2n ) ∈ (P1 :R1 M1 ) × R2 = (P1 × M2 :R1 ×R2
M1 × M2 ) for some positive integer n ∈ N or (m1 , m2 ) ∈ P1 × M2 . So P1 × M2 is
an I-primary submodule of M1 × M2 .
4.
The proof is similar to part (2).
5.
The proof is similar to part (3).

Proposition 2.10. Let M be an R-module and let P be a proper submodule of M. Then
M
P
is 0-primary in IP
.
P is I-primary in M if and only if IP

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5

Proof. Suppose that P is I-primary in M. Let r ∈ R, m ∈ M such that 0 , rm + IP =
P
M
r(m + IP) ∈ IP
in IP
. Then rm ∈ P − IP. But by assuming P is I-primary, so m ∈ P
q
√
√
M
P
P
P
or r ∈ (P : M). Then r ∈ P : M = ( IP
: IP
) or m + IP ∈ IP
. Hence IP
is 0-

P
M
M
. Conversely suppose that IP
is 0-primary submodule in IP
.
primary submodule in IP
P
P
Let r ∈ R, m ∈ M such that rm ∈ P − IP. So 0 , r(m + IP) = rm + IP ∈ IP . But IP is
q
√
M
P
M
P
{0}-primary in IP
. Thus m + IP ∈ IP
or r ∈ ( IP
: IP
) and so m ∈ P or r ∈ (P : M).
Hence P is 0-primary.


Now we give two other charactrizations of I-primary submodules.
Theorem 2.11. Let M be an R-module and P be a proper submodule of M. Then P is
I-primary in M if and only √if for any ideal J of R and submodule N of M such that
JN ⊆ P − IP, we have J ⊆ (P : M) or N ⊆ P
Proof. Suppose that P is I-primary submodule√ of M, and JN ⊆ P − IP for some
ideal J of
√ R and submodule N of M. If J * (P : M) and N * P, so there exists
r ∈ J − (P : M) and x ∈ N
√ − P such that rx ∈ P − IP. By assuming P is I-primary
√
in M, either x ∈ P or r ∈ (P : M) which is a contradiction. Hence J ⊆ (P : M)
or N ⊆ P. Conversely suppose that rm ∈ P − IP for r ∈ √R and m ∈ M. Then
(r)(m) = (rm)√⊆ P − IP. So by assumption, either (r) ⊆ (P : M) or (m) ⊆ P.
Therefore r ∈ (P : M) or m ∈ P. Thus P is an I-primary submodule of M.

The following theorem gives the relation between I-primary ideals and I-primary
submodules.
Theorem 2.12. Let M be a finitely generated faithful multiplication R-module and P a
proper submodule of M with (IP : M) = I(P : M). Then P is an I-primary submodule
of M if and only if (P : M) is an I-primary ideal of R.
Proof. Assume that P is I-primary submodule in M. Let r, s ∈ R, such that
rs ∈ (P : M) − I(P : M). Then rsM ⊆ P. If rsM ⊆ IP. Then rs ∈ (IP : M) =√I(P : M)
which is contradiction. Assume
√ rsM * IP. Then rsM ⊆ P − IP. So r ∈ (P : M)
or sM ⊆ P. Therefore r ∈ (P : M) or s ∈ (P : M). Hence (P : M) is I-primary
ideal. Conversely assume that (P : M) is I-primary ideal. Let r ∈ R, m ∈ M such that
rm ∈ P − IP. r(Rm : M) = (rRm : M) ⊆ (P : M) and r(Rm : M) * I(P : M) otherwise
rRm = r(Rm : √
M)M ⊆ I(P : M)M = IP. That is r(Rm : M) ⊆√(P : M) − I(P : M).
Therefore√r ∈ (P : M) or (Rm : M) ⊆ (P : M). Hence r ∈ (P : M) or Rm ⊆ P.
Thus r ∈ (P : M) or m ∈ P. This means that P is I-primary submodule.

Theorem 2.13. Let M be a finitely generated faithful multiplication R-module and P
be a proper submodule of M such that I(P : M) = (IP : M). Then P is I-primary in
M if and only if whenever N
√ and K are submodules of M such that NK ⊆ P − IP, we
have either N ⊆ P or K ⊆ P.

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I. Akray and H. S. Hussein

Proof. Suppose that P is an I-primary submodule in M. So (N : P) is an I-primary
ideal in R. As M is a multiplication R-module, N = (N : M)M, K = (K : M)M √
and
so N.K = (N : M)(K : M)M. Suppose that
NK
⊆
P
−
IP,
but
N
*
P
and
K
*
P.
√
Thus (N : M) * (P : M) and (K : M) * (P : M). As (P : M) is I-primary in R. So
either (N : M)(K : M) * (P : M) or (N : M)(K : M) ⊆ I(P : M). In the first case,
we have N.K = (N : M)(K : M)M ⊆ (P : M)M = P which is contradiction. Now if
(N : M)(K : M) ⊆ I(P : M), then N.K = (N : M)(K : M)M ⊆ I(P : M)M = IP,
which is again a contradiction. Hence either N ⊆ P or K ⊆ P. To prove the converse
part, by Theorem 2.12, it is enough to prove that (P : M) is I-primary ideal in R. Let
a, b ∈ R such that ab ∈ (P : M) − I(P : M) with a < (P : M) and b < (P : M). Take
N = aM, K = bM. Then N.K = abM ⊆ (P : M)M = P. If NK = abM ⊆ IP, then
ab ∈ (IP : M) = I(P : M), which is contradiction. Hence NK ⊆ P − IP. By hypothesis
and since M √
is a faithful
multiplication module, we have either
√
√ N = aM ⊆ P or
K = bM ⊆ P = (P : M)M and so a ∈ (P : M) or b ∈ (P : M). Therefore
(P : M) is an I-primary ideal of R.

Proposition 2.14. Let P be a submodule of an R-module M and N be any R-module.
Then P is an I-primary submodule of M if and only if P ⊕ N is an I-primary submodule
of M ⊕ N.
Proof. Assume that P is an I-primary submodule in M. Let r ∈ R, (m, n) ∈ M ⊕ N
such that r(m, n)√∈ (P ⊕ N) − I(P ⊕ N). Then rm ∈ P −√IP. But P is I-primary in M,
so m ∈ P or r ∈ (P : M), that is (m, n) ∈ P ⊕ N or r ∈ (P ⊕ N : M ⊕ N). Therefore
P ⊕ M is I-primary in M ⊕ N. Conversely suppose that P ⊕ N is I-primary in M ⊕ N.
Let r ∈ R, m ∈ M such that rm ∈ P − IP. Then r(m, 0) ∈√ (P ⊕ N) − I(P ⊕ N). But by
assuming P ⊕√N is I-primary, so (m, 0) ∈ P ⊕ N or r ∈ (P ⊕ N : M ⊕ N). Therefore
m ∈ P or r ∈ (P : M). Hence P is I-primary in M.

Let J be an ideal. We show that under a certain condition the I-primaryness of
submodules P and (P : J) are equivalent. First we give the following lemma.
Lemma 2.15. Let P be a submodule of a faithful multiplication R-module M and J be
finitely generated faithful multiplication ideal of R, then:
(1) P = (JP : J);
(2) If P ⊆ JM, then (KP : J) = K(P : J) for any ideal K of R.
Proposition 2.16. Let P be a submodule of a faithful multiplication R-module M and
J be a finitely generated faithful multiplication ideal of R. Then P is an I-primary
submodule of JM if and only if (P : J) is I-primary in M.
Proof. Suppose P is I-primary in JM. Take r ∈ R, m ∈ M such that rm ∈ (P : J)−I(P :
J). Then rJm ⊆ P − IP. If rJm * IP, then by Lemma 2.15 , rm ∈ (IP : J)√= I(P : J)
which is a contradiction. √As P is an I-primary in JM, then Jm ⊆ P or r ∈ (P : JM).
So m ∈ (P : J) or r ∈ ((P : J) : M), because (P : JM) = ((P : J) : M). Hence
(P : J) is an I-primary submodule in M. Conversely, assume that (P : J) is I-primary

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7

in M. Let K be an ideal of R, and N be a submodule of JM such that KN ⊆ P − IP.
Then K(N : J) ⊆ (KN : J) ⊆ (P : J). Moreover, if K(N : J) ⊆ I(P : J), then
by Lemma 2.15 KN = K(JN : J) = K(N : J)J ⊆ I(P : J)J = IP which is a
contradiction.√ Hence K(N : J)√⊆ (P : J) − I(P : J). As (P : J) is I-primary in M,
either K ⊆ ((P : J) : M) = (P : JM) or (N : J) ⊆ (P : J), which implies that
N = (N : J)J ⊆ (P : J)P = P. Hence P is an I-primary submodule in JM.

Let M and F be R-modules and r ∈ R. Then it is clear that for any submodule
P of M, F ⊗ (P : r) ⊆ (F ⊗ P : r). In the following lemma we give a condition under
which the equality holds.
Lemma 2.17. Let r ∈ R and P a submodule of M. Then for every flat R-module F, we
have F ⊗ (P : r) = (F ⊗ P : r).
fr

Proof. Consider the exact sequence 0 −→ (P : r) −→ M −→ MP where fr (m) = rm+P.
As F is flat , the exactness of the sequence 0 −→ P −→ M −→ MP −→ 0 implies the
exactness of the sequence 0 −→ F ⊗ P −→ F ⊗ M −→ F ⊗ MP −→ 0 which gives
M
F ⊗ MP  F⊗M
F⊗P . So the exactness of the sequence 0 −→ (P : r) −→ M −→ P imply the
1⊗ f´r

´
exactness of the sequence 0 −→ F ⊗(P : r) −→ F ⊗ M −→ F⊗M
F⊗P where (1⊗ fr )(n⊗m) =
´
r.(n ⊗ m) + F ⊗ P for n ∈ F. Therefore F ⊗ (P : r) = ker(1 ⊗ fr ) = (F ⊗ P :F⊗M r). 
Theorem 2.18. Let P be an I-primary submodule of an R-module M and F be a flat
R-module with F ⊗ P , F ⊗ M. Then F ⊗ P is an I-primary submodule of F ⊗ M.
√
Proof. Suppose that P is an I-primary and r ∈ R − (P : M). Then by Theorem 2.4
(P : r) = P or (P : r) = (IP : r). So by Lemma 2.17, (F ⊗ P : r) = F ⊗ (P : r) = F ⊗ P
or (F ⊗ P : r) = F ⊗ (P : r) = F ⊗ (IP : r) = (F ⊗ IP : r) = (I(F ⊗ P) : r) and
consequently F ⊗ P is an I-primary submodule of F ⊗ M.

Proposition 2.19. Let F be a faithfully flat R-module. Then a submodule P of an Rmodule M is I-primary if and only if F ⊗ P is an I-primary submodule of F ⊗ M.
Proof. Suppose that P is an I-primary submodule of an R-module M and F a
faithfully flat R-module. If F ⊗ P = F ⊗ M, then the exactness of the sequence
0 −→ F ⊗ P −→ F ⊗ M −→ 0 imply the exactness of 0 −→ P −→ M −→ 0 and
hence P = M which is a contradiction. So F ⊗ P , F ⊗ M and by Theorem 2.18 F ⊗ P
is an I-primary submodule of F ⊗ M. Conversely, let F ⊗ P be an I-primary submodule
of F ⊗ M.
√ Hence we obtain F ⊗ P , F ⊗ M and so P , M. By Lemma 2.17, for every
r ∈ R − (P : M) we have r < (F ⊗ P : F ⊗ M) and F ⊗ (P : r) = (F ⊗ P : r) = F ⊗ P
or F ⊗ (P : r) = (F ⊗ P : r) = (I(F ⊗ P) : r) = (F ⊗ IP : r) = F ⊗ (IP : r). Suppose
F ⊗ (P : r) = F ⊗ P. Then 0 −→ F ⊗ (P : r) −→ F ⊗ P −→ 0 is an exact sequence and
as F is faithfully flat, 0 −→ (P : r) −→ P −→ 0 is an exact sequence and consequently
(P : r) = P. The other case can be proved similarly. Thus by Theorem 2.4, P is an
I-primary submodule of M.


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I. Akray and H. S. Hussein

It is well-known that I ⊗ F  IF for any ideal I of R and flat R-module F. Thus by
Theorem 2.18 and Proposition 2.19 we conclude the following.
Corollary 2.20. Let F be a flat R-module and P be an I-primary ideal of R with
PF , F. Then PF is an I-primary submodule of F. In the case F is faithfully flat, the
converse is also true.
We know that every polynomial ring R[x] is a flat R-module and that R[x] ⊗ M 
M[x]. Hence as an immediate consequence of the Theorem 2.18 we give the following
corollary.
Corollary 2.21. Let M be an R-module and x an indeterminate. If P is an I-primary
submodule of M, then P[x] is an I-primary submodule of M[x].

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Ismael Akray, Department of Mathematics, Soran University, Erbil city, Kurdistan
region, Iraq
e-mail: ismael.akray@soran.edu.iq
Halgurd S. Hussein, Department of Mathematics, Soran University, Erbil city, Kurdistan region, Iraq
e-mail: rasty.rosty1@gmail.com

