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Properties of the H-Supplemented Modules Relative to the Class of B(M,X)

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Yahya Talebi at University of Mazandaran
Yahya Talebi
Mehrab Hosseinpour at University of Mazandaran
Mehrab Hosseinpour
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Abstract

Let M be a right R-module. M is called X-H-supplemented if for every submodule A of M with A ∈ B(M;X) there exists a direct summand D of M such that M = A + Y if and only if M = D + Y for all Y ⊆ M. In this paper several properties of such modules are studied and we show that if M is a UCC-module, then M is X-H-supplemented if and only if M is X-lifting.
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Journal of Advanced Research in
Pure Mathematics
Online ISSN: 1943-2380
Vol. 5, Issue. 2, 2013, pp. 18-24
doi: 10.5373/jarpm.1325.022112
Properties of the H-supplemented modules relative to
the class of B(M, X)
Yahya Talebi
∗
, Mehrab Hosseinpour
Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran,
Babolsar, Iran
Abstract. Let M be a right R-module. M is called X-H-supplemented if for every
submodule A of M with A ∈ B(M, X) there exists a direct summand D of M such
that M = A + Y if and only if M = D + Y for all Y ⊆ M. In this paper several
properties of such modules are studied and we show that if M is a U CC-module, then
M is X-H-supplemented if and only if M is X-lifting.
Keywords: H-supplemented module; X-H-supplemented module.
Mathematics Subject Classification 2010: 16D60, 16D99.
1 Introduction
Throughout this paper R is an associative ring with unity and all modules are unitary
right R-modules. A submodule K of a module M is denoted by K ≤ M. The notation
N ≤
⊕
M, means that N is a direct summand of M. A submodule N of M is called
small in M, denoted by N ≪ M, if N +L ̸= M for every proper submodule L of M. Let
M be an R-module and N ⊆ K ⊆ M. N is said to be a cosmall submodule of K in M
if K/N ≪ M/N (denoted by N ≤
cs
K). A submodule N of M is coclosed in M if it has
no proper cosmall submodules in M (denoted by N ≤
cc
M). N is called a coclosure of K
in M, if N ≤
cs
K in M and N ≤
cc
M. A module M is called a unique coclosure module
(UCC-module), if every submodule of M has a unique coclosure in M. If any submodule
K of M is minimal with the prop erty that M = N + K, then the submodule K is called
a supplement of N in M. It is easy to see that K is a supplement of N in M if and
only if M = N + K and N ∩ K ≪ K. Following [9], M is called supplemented if every
submodule of M has a supplement in M. Following [7], M is called ⊕-supplemented if
∗
Correspondence to: Yahya Talebi, Department of Mathematics, Faculty of Mathematical Sciences,
University of Mazandaran, Babolsar, Iran. Email: talebi@umz.ac.ir
†
Received: 21 February 2012, revised: 8 May 2012, accepted: 14 August 2012.
http://www.i-asr.com/Journals/jarpm/ 18
c
⃝2013 Institute of Advanced Scientific Research
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