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Journal of Discrete Mathematical Sciences and
Cryptography
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Small prime and quasi-small prime modules
Wisam A. Ali & Nuhad S. Al. Mothafar
To cite this article: Wisam A. Ali & Nuhad S. Al. Mothafar (2021) Small prime and quasi-small
prime modules, Journal of Discrete Mathematical Sciences and Cryptography, 24:7, 1967-1971,
DOI: 10.1080/09720529.2021.1961889
To link to this article: https://doi.org/10.1080/09720529.2021.1961889
Published online: 25 Oct 2021.
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Small prime and quasi-small prime modules
Wisam A. Ali *
Nuhad S. Al. Mothafar †
Department of Mathematics
College of Science
University of Baghdad
Baghdad
Iraq
Abstract
Let R be a commutative ring with identity and W be a unitary ( left ) R- module. In
this paper, we introduce and study the relation between small prime and quasi-small prime
modules.
Subject Classification: Primary 08C15, Secondary 30D60.
Keywords: Small submodules, Small prime modules, Quasi – small prime modules.
1. Introduction
Throughout this research, R is a commutative ring with identity, and
W is a unitary (left) R- module. Mahmood, L. S. in 2012, [4] introduced
the concept of small prime iall non- zero small submodule H of W. Where
a submodule H of W is called small ( notational,
)H W
if H + L= W
for all submodule L of W implies L = W, [2]. In 2021, were introduced
the concept of a quasi – small prime modules, [1]. Where they call W is a
quasi – small prime module if and only if annR H is a prime ideal for all
non-zero small submodule H of W. Recall that an R-module W is called
faithful if annR W = (0), [ 6 ]. An R-module W is called multiplication if
for all submodule H of W, there exist an ideal I of R such that IW = H, in
particular
= :
[ ] ,[6].
R
H HWW
*E-mail: wisam.azizali1203@sc.uobaghdad.edu.iq (Corresponding Author)
†E-mail: nuhad.salim@sc.uobaghdad.edu.iq
Journal of Discrete Mathematical Sciences & Cr yptography
ISSN 0972-0529 (Print), ISSN 2169-0065 (Online)
Vol. 24 (2021), No. 7, pp. 1967–1971
DOI : 10.1080/09720529.2021.1961889
1968 W. A. ALI AND N. S. A. MOTHAFAR
2. The Relation Between Small Prime and Quasi-Small Prime
Modules
In this section, we introduce and study the relation between small
prime and quasi – small prime modules.
Recall that an R-module W is called a small prime if and only if
=
RR
ann W ann H
for all non-zero small submodule H of W. Equivantly,
an R-module W is called small prime if and only if (0) is a small prime
submodule of W, [4].
We know that, if W is a quasi- small prime R-module, then
R
ann W
need not be small prime ideal. For example: the Z4-module Z4 is a quasi-
small prime, but
=
444
4
z
ann Z Z
is not a small prime ideal of Z4 , since
´Î
4
2 2 4Z
but
Ï4
24.Z
Proposition 2.1 : Let W be a finitely generated faithful multiplication
R-module. If Wis a quasi- small prime R-module, then annRW is a small
prime ideal of R.
Proof : Let
ÎÎ, ,( ) and .
R
a b R b R ab ann W
Then
=( ) (0).ab W
But
Í( ) () ,ab b R
implies
() ,ab R
[3]. Since W is a finitely generated
faithful multiplication R-module, then
( ) , [3].ab W W
Since W is a
quasi- small prime R-module, therefore, either = 0 or bW = 0. Hence either
ÎÎ or .
RR
a ann W b ann W
Thus annRW is a small prime ideal of R.
Lemma 2.2 : Let W be a finitely generated faithful multiplication R-module. If
W is a quasi- small prime R-module, then (0) is a small prime submodule of W.
Proof : Let
ÎÎ,a Rw W
with
()wW
such that aw = 0, then a(w) = 0.
Since W is a multiplication R- module, then there exists an ideal I of R
such that (w) = IW. Hence aIW = (0),which implies that
Í.
R
aI ann W
But
annRW is a small prime ideal of R by proposition (2.1) and
,IR
so either
Î
R
a ann W
or
Í.
R
I ann W
Therefore either
Î
R
a ann W
or
=(0),IW
hence
either
Î
R
a ann W
or (w) = (0). Thus (0) is a small prime submodule of W.
Theorem 2.3 : Every finitely generated faithful multiplication quasi-small prime
R-module is a small prime.
Proof : Let W is a quasi-small prime f.g. faithful multiplication R-module.
Since (0) is a small prime submodule of W by lemma (2.2), then W is a
small prime R-module, [4].
Corollary 2.4 : If W is a finitely generated faithful multiplication R-module, then
W is a small prime R-module if and only if W is a quasi-small prime R-module.
Corollary 2.5 : If W is a cyclic faithful quasi-small prime R-module, then W is a
small prime R-module.
SMALL PRIME AND QUASI-SMALL 1969
Proof : Since W is cyclic, then it is clear that W is a multiplication. Hence,
W is a small prime R-module by corollary (2.4).
Recall that an R-module W is called a quasi – Dedekind if Hom(W⁄H,W)
= 0 for all non – zero submodule H of W, [5] .
Proposition 2.6 : If Wis quasi-Dedekind R-module, then W is quasi-small
prime module.
Proof : Since W is a prime R-module, [5]. Then W is a quasi-small prime
R-module, [1].
Recall that an R-module W is said to be chained module if and only if
every non-empty set of submodules are ordered by inclusion [2].
Notation: A cyclic submodule H of an R-module W is called a small cyclic
(for short s-cyclic) submodule if it is small in W.
An R-module W is called s-cyclic if every cyclic submodule of Wis
small.
Proposition 2.7 : If W is a quasi-small prime s-cyclic and chained R-module,
then W is a small prime R-module.
Proof : Let (w) be a non-zero small submodule of W, we have to show
that
=( ),
RR
ann W ann w
[4]. It is clear that
Í( ).
RR
ann W ann w
Let
Î ( ),
R
a ann w
so aw = 0. Suppose that
Ï,
R
a ann W
hence there exists
ι
11
such that 0, w W aw
since W is s-cyclic, then
1
()wW
such that
¹
1
0.aw
But W is chained R-module, so either
Í
1
() ()ww
or
Í
1
( ) ( ).ww
If
Í
1
( ) ( ),ww
then
Í
1
() ( )
RR
ann w ann w
so
=
10,aw
which is contradiction.
Thus,
Í
1
() ()ww
and hence w = bw1 for some
ÎbR
and
¹0.b
==
1
0,aw abw
implies that
Î
1
().
R
a ann bw
But W is a quasi-small prime
R-module and
¹
10,bw
1
() .wW
Hence by [1],
=
11
() ,()
RR
ann w ann bw
Î
1
( ),
R
a ann w
so
=
10,aw
which is a contradiction. Thus
Î.
R
a ann W
Therefore,
Í()
RR
ann w ann W
and hence
=( ).
RR
ann W ann w
Then W is a
small prime R-module, [4].
Corollary 2.8 : If W is s-cyclic and chaine R-module, then W is a small prime
R-module if and only if W is a quasi-small prime R-module.
Corollary 2.9 : Every chaine s-cyclic quasi- small prime R-module is a quasi –
Dedekind.
Proof : Let W be a module. By proposition (2.7), W is a small prime a
module Since W is s-cyclic, then W is a prime R-module , [4]. But W is
chained, so by [5] the result is follows.
Recall that an R-module W is called a hollow module if every proper
submodule of W is small, [2].
1970 W. A. ALI AND N. S. A. MOTHAFAR
Corollary 2.10 : Let W be a chaine hollow R-module. Then the following
statements are equivalents:
1. W is a quasi – small prime R- module.
2. W is a small prime R- module.
3. W is a prime R – module.
4. W is a quasi – Dedekind R- module.
Proof: (1) (2) By proposition (2.7).
(2) (3) Since W is hollow, then W is a prime R- module, [4].
(3) (4) By [5].
(4) (1) By proposition (2.6).
Recall that an R-module W is called uniform if every non-zero
submodule of W is essential, [2].
Proposition 2.11 : If W is a uniform s-cyclic and quasi-small prime
R-module. Then W is a small prime R-module.
Proof : To show that
=()
RR
ann W ann w
for all non-zero small submodule
(w) of W. Let
Î()
R
a ann w
so aw = 0. Let
¹Î
1
0,wW
since W is s-cyclic,
then,
1
() .wW
Since W is uniform, then
ǹ
1
( ) ( ) (0)ww
and so there
exist
Î and b cR
such that
=¹
1
0.bw cw
But aw = 0, implies that abw =
0. It follows that
==
10.abw acw
Hence
Î
1
( ),
R
a ann cw
but W is a quasi-
small prime R-module, therefore,
=
11
() ,()
RR
ann w ann cw
Î
1
( ),
R
a ann w
[1],
which mean
=
1
0aw
for all
Î
1
.wW
This implies that
Î
R
a ann W
and so
=( ).
RR
ann W ann w
Therefore, W is a small prime R-module, [4].
Corollary 2.12 : If W is a uniform and s-cyclic R-module, then W is a small
prime R-module if and only if W is a quasi-small prime R-module.
Corollary 2.13 : Every uniform s-cyclic quasi- small prime module is a quasi –
Dedekind.
Proof : Let W be an R-module. By proposition (2.11), W is a small prime
R- module. Since W is s - cyclic, then W is a prime R- module, [4]. But W is
uniform, so by [5] the result is follows.
Corollary 2.14 : Let W be a uniform hollow R-module. Then the following
Statements are equivalents:
1. W is a quasi – small prime R- module.
2. W is a small prime R- module.
3. W is a prime R- module.
4. W is a quasi – Dedekind R- module.
SMALL PRIME AND QUASI-SMALL 1971
Proof: (1) (2) By proposition (2.11).
(2) (3) Since W is hollow, then W is a prime R-module, [4].
(3) (4) By [5].
(4) (1) By proposition (2.6).
References
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Modules”, Journal of Physics : Conference Series, 1818, 01204 .
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Modules”.
[3] Athab, I. A, (2004), “Some Generalization of Projective Modules”, Ph.
D. Thesis, College of Science, University of Baghdad.
[4] Mahmood, L. S. (2012), “Small Prime Modules and Small Prime
Submodules”, Journal of Al-Nahrain University, 15(4), PP:191-199.
[5] Mijbass, A. S., (1997), “Quasi-Dedekined Modules and Quasi-
Invertible Submodules”,
[6] Ph. D. Thesis, College of Science, Univ. of Baghdad.
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Received May, 2021
Revised June, 2021