On τ-Extending Modules

Abstract

Motivated by [2] and [6], we introduce a generalization of extending (CS) modules by using the concept of τ -large submodule which was defined in [9]. We give some properties of this class of modules and study their relationship with the familiar concepts of τ -closed, τ -complement submodules and the other generalization of extending modules (τ -complemented, τ -CS, s-τ -CS modules). We are also interested in determining when a τ -divisible module is τ -extending. For a τ -extending module M with C3, we obtain a decomposition theorem that there is a submodule K of M such that M = τ (M ) ⊕ K and K is τ (M )-injective. We also treat when a direct sum of τ -extending modules is τ -extending.Mathematics Subject Classification (2010). Primary 16S90; Secondary 16D50, 16D70.

The Osofsky-Smith Theorem for Modular Lattices, and Applications (II)

“…The module R R in Example 5.5 is -CS but not strongly -CS because is -(essentially) closed but not a direct summand of R. Note that the authors use the confusing related terms of a -complement submodule of a module as given in Gómez Pardo [23] and that of a -complemented module as given in Smith et al [29]. Also, note that most of the results of Section 2 in Çeken and Alkan [16], e.g., their Proposition 2. We say that a finite family N i 1 i n of submodules of a module M R isindependent if N i ∈ ∀ i 1 i n, and…”

“…Note that for any hereditary torsion theory = , the -extending modules are exactly the -complemented modules in the sense of Smith et al [29] we discussed and called strongly -complemented. (9) 2012: Çeken and Alkan [16] use the concepts of -essential, -complement, and -essentially closed as introduced by Gómez Pardo [23] in order to define the relative concept of a -CS module: a module M R is called -CS if every -(essentially) closed submodule of M is a direct summand of M. Because, as we will see right away, this concept is different of ours, we will call such modules strongly -CS. The module R R in Example 5.5 is -CS but not strongly -CS because is -(essentially) closed but not a direct summand of R. Note that the authors use the confusing related terms of a -complement submodule of a module as given in Gómez Pardo [23] and that of a -complemented module as given in Smith et al [29].…”

“…The author thanks N. V. Dung for drawing to his attention the paper Dung [19], Constantin Nȃstȃsescu and John Van Den Berg for helpful discussions, Septimiu Crivei for providing him references [13,14,16], and the referee for his/her careful reading of the manuscript and helpful suggestions. …”

On $\tau$-extending modules

“…A module M is extending if every submodule of M is essential in a direct summand of M . In recent years, torsion-theoretic analogues of extending modules have been studied by many authors (see [4], [15], [5], [7], [16], [10], [8]).…”

“…In [4] the authors say that M is τ -extending module if every submodule is τ -large in a direct summand of M . They showed that every τ -torsion module is τ -extending and they also proved that a τ -torsion free module is τ -extending if and only if it is extending.…”