Hindawi Publishing Corporation
Te Scientifc World Journal
Volume 2013, Article ID 750808, 3 pages
http://dx.doi.org/10.1155/2013/750808
Research Article
The Unique Maximal GF -Regular Submodule of a Module
Areej M. Abduldaim and Sheng Chen
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
Correspondence should be addressed to Areej M. Abduldaim; areej mussab@yahoo.com
Received 20 July 2013; Accepted 15 August 2013
Academic Editors: J. He, I. G. Ivanov, J. Rada, and L. Sz´ ekelyhidi
Copyright © 2013 A. M. Abduldaim and S. Chen. Tis is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
An -module is called -regular if, for each ∈ and ∈, there exist ∈ and a positive integer such that
=
.
We proved that each unitary -module contains a unique maximal -regular submodule, which we denoted by ().
Furthermore, the radical properties of are investigated; we proved that if is an -module and is a submodule of , then
()=∩(). Moreover, if is projective, then () is a -pure submodule of and ()=()⋅.
1. Introduction
Troughout this paper, is a commutative ring with identity
and all modules are lef unitary, unless otherwise stated.
Recall that an element ∈ is said to be regular if there exists
∈ such that =; a ring is called regular if and only if
each element of is regular. An ideal of a ring is regular
if each of its elements is regular in ; indeed, a regular ideal
of is itself a regular ring [1]. Brown and McCoy proved in
[1] that each ring contains a unique maximal regular ideal
() which satisfes the well-known radical properties. Te
ideal () is called the regular radical of .
Te concept of regularity was extended to modules in
several ways and in [2] the notion of -regular modules (in
the sense of Fieldhouse [3]) was generalized to -regular
modules. Let be an -module; an element ∈ is said
to be -regular if for each ∈ there exist ∈ and
a positive integer such that
=
. An -module
is called -regular if and only if all its elements are -
regular; in particular, a ring is -regular if and only if
is -regular as an -module. On the other hand a ring
is -regular if and only if is a -regular -module; recall
that a ring is -regular if, for each ∈, there exist ∈
and a positive integer such that
=
. A submodule
of an -module is called -regular if each element of
is -regular and every submodule of a -regular module
is a -regular module. Also, in [2] the concept of -pure
submodules was introduced; a submodule of an -module
is called -pure if, for each ∈, there exists a positive
integer such that ∩
=
.
In this paper we show that each module contains a
unique maximal -regular submodule, which we denote by
(), and we show that () satisfes some but not
all of the usual radical properties.
2. Main Results
Teorem 1. Let be any ring. Every -module contains a
unique maximal -regular submodule.
Proof. Let be any ring, let be an -module, and let
={| is a -regular submodule of }, (1)
where ̸ = because (0) is a -regular submodule of . Let
{
} be an ascending chain in and =⋃
∈Λ
. Let ∈;
there exists ∈Λ such that ∈
, but
is a -regular
submodule; then; for each ∈, there exist ∈ and a
positive integer such that
=
; therefore is a -
regular element in which implies that is a -regular
-module. Now, by Zorn’s lemma, contains a maximal
element which we call . To prove the uniqueness of
, assume that 1 and 2 be two maximal -
regular submodules in ; then for any maximal ideal of
each of 1
and 2
is semisimple over
[2,
Proposition 21]. Now, let 1
∩2
=
; then
⊆
1
and
⊆ 2
; thus 1
=
+1
and