On the Prime Submodules of Multiplication Modules
نویسنده
چکیده
By considering the notion of multiplication modules over a commutative ring with identity, first we introduce the notion product of two submodules of such modules. Then we use this notion to characterize the prime submodules of a multiplication module. Finally, we state and prove a version of Nakayama lemma for multiplication modules and find some related basic results. 1. Introduction. Let R be a commutative ring with identity and let M be a unitary R-module. Then, M is called a multiplication R-module provided for each submodule N of M; there exists an ideal I of R such that N = IM. Note that our definition agrees with that of [1, 2], but in [6] the term multiplication module is used in a different way. (In this paper, an R-module M is a multiplication if and only if every submodule of M is a multiplication module in the above sense.) Recently, prime submodules have been studied in a number of papers; for example, see [3, 4, 5]. Now in this paper, first we define the notion of product of two submodules of a multiplication module and then we obtain some related results. In particular, we give some equivalent conditions for prime submodules of multiplication submodules. Finally, we state and prove a version of Nakayama lemma for multiplication modules.
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