On some properties of ⊕‐supplemented modules

Abstract

A module M is ⊕-supplemented if every submodule of M has a supplement which is a direct summand of M. In this paper, we show that a quotient of a ⊕-supplemented module is not in general ⊕-supplemented. We prove that over a commutative ring R, every finitely generated ⊕-supplemented R-module M having dual Goldie dimension less than or equal to three is a direct sum of local modules. It is also shown that a ring R is semisimple if and only if the class of ⊕-supplemented R-modules coincides with the class of injective R-modules. The structure of ⊕-supplemented modules over a commutative principal ideal ring is completely determined.2000 Mathematics Subject Classification: 16D50, 16D60, 13E05, 13E15, 16L60, 16P20, 16D80. Introduction.All rings considered in this paper will be associative with an identity element. Unless otherwise mentioned, all modules will be left unitary modules. Let R be a ring and M an R-module. Let A and P be submodules of M. The submodule P is called a supplement of A if it is minimal with respect to the property A + P = M. Any L ≤ M which is the supplement of an N ≤ M will be called a supplement submodule of M. If every submodule U of M has a supplement in M, we call M complemented. In [25, page 331], Zöschinger shows that over a discrete valuation ring R, every complemented R-module satisfies the following property (P ): every submodule has a supplement which is a direct summand. He also remarked in [25, page 333] that every module of the form M (R/a 1 ) × ··· × (R/a n ), where R is a commutative local ring and a i (1 ≤ i ≤ n) are ideals of R, satisfies (P ). In [12, page 95], Mohamed and Müller called a module ⊕-supplemented if it satisfies property (P ).On the other hand, let U and V be submodules of a module M. The submodule V is called a complement of U in M if V is maximal with respect to the property V ∩U = 0. In [17] Smith and Tercan investigate the following property which they called (C 11 ): every submodule of M has a complement which is a direct summand of M. So, it was natural to introduce a dual notion of (C 11 ) which we called (D 11 ) (see [6,7]). It turns out that modules satisfying (D 11 ) are exactly the ⊕-supplemented modules. A module M is called a completely ⊕-supplemented (see [5]) (or satisfies (D

On generalization of ⊕-cofinitely supplemented modules

“…By Proposition 2.1, M is˚-(cofinitely) supplemented. This is a contradiction according to [10] (Example 2.3).…”

Supplemented Modules Relative to an Ideal

“…(ii) Note that [11,Corollary 4.5] shows that every ⊕-supplemented R-module is completely ⊕supplemented. Our next goal is to describe ⊕-δ-supplemented modules and I-⊕-supplemented modules over a nonlocal Dedekind domain R. The next proposition shows that every torsion-free δ-supplemented R-module is injective.…”

Weakly distributive modules. Applications to supplement submodules

“…Factor modules of ⊕-supplemented modules need not be ⊕-supplemented in general (see Example 2.2 of [5]). When the module is weakly distributive we have the following.…”