Abstract
Let R be a commutative ring with unity and M is a unitary left R-module . In this paper , we introduce the notion of strongly S-coprime modules, where M is called strongly S-coprime briefly (SS-Coprime) if for each r R , r2M is small in M implies rM=0 . We investigate many properties about this concept. Moreover many relations between it and other related concepts, are given. Â Â Â
Highlights
Let R be a commutative ring with unity and let M be a unital R-module
Recall that a proper submodule N of M is called prime if whenever r ∈ R, x ∈ M, rx ∈ N implies that x ∈ N or r ∈ [N: M],[10]
Yassemi in [13] introduced the notions of second submodule and second module, where a submodule N of M is called second if whenever r∈R-{0}, rN=N or rN=0
Summary
Let R be a commutative ring with unity and let M be a unital R-module. Recall that a proper submodule N of M is called prime if whenever r ∈ R , x ∈ M , rx ∈ N implies that x ∈ N or r ∈ [N: M],[10]. S. Annine in [2] introduced the notion of coprime module as follows: M is called a coprime R-module if annM=annMN for each proper submodule N of M. It is clear that T-noncosingular module is S-coprime and the converse is not true in general, as we shall see later. In section two of this paper we investigate the notion of strongly S-coprime module (briefly SS-coprime) where an Rmodule M is called SS-coprime if for any a,b∈R, abM
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