Let R be a commutative ring with identity. Let N and K be two submodules of a multiplication R-module M. Then N=IM and K=JM for some ideals I and J of R. The product of N and K denoted by NK is defined by NK=IJM. In this paper we characterize some particular cases of multiplication modules by using the product of submodules.

primary-like and weakly primary-like submodules are two new generalizations of primary ideals from rings to modules. in fact, the class of primary-like submodules of a module lie between primary submodules and weakly primary-like submodules properly.  in this note, we show that these three classes coincide when their elements are submodules of a multiplication module and satisfy the primeful pr...

Let R be a commutative ring with unity. And let E unitary R-module. This paper introduces the notion of 2-prime submodules as generalized concept ideal, where proper submodule H module F over is said to if , for r and x implies that or . we prove many properties this kind submodules, then only [N ] E, R. Also, non-zero multiplication module, [K: F] [H: every k such K. Furthermore, will study ba...

Let $R\ $be a commutative ring with $1\neq0$ and $M$ be an $R$-module. Suppose that $S\subseteq R\ $is multiplicatively closed set of $R.\ $Recently Sevim et al. in \cite{SenArTeKo} introduced the notion $S$-prime submodule which is generalization prime used them to characterize certain classes rings/modules such as submodules, simple modules, torsion free modules,\ $S$-Noetherian modules etc. ...

Primary-like and weakly primary-like submodules are two new generalizations of primary ideals from rings to modules. In fact, the class of primary-like submodules of a module lie between primary submodules and weakly primary-like submodules properly.  In this note, we show that these three classes coincide when their elements are submodules of a multiplication module and satisfy the primeful pr...

The purpose of this paper is to explore some basic facts from radical of submodules in the free multiplication R-module M = Rn. Mathematics Subject Classification: 13C13, 13C99

We introduce the concept of "uniformly primary submodules" of a module over a commutative ring R, which generalizes the concept of "uniformly primary ideals" of R, a concept that imposes a certain boundedness condition on the usual notion of "primary ideal". Several results on uniformly primary submodules are proved. Also, we characterize uniformly primary submodules of a multiplication module....

Let A be a finite dimensional Hopf algebra, D(A) = A ⊗A its Drinfel’d double and H(A) = A#A its Heisenberg double. The relation between D(A) and H(A) has been found by J.-H. Lu in [24] (see also [33], p. 196): the multiplication of H(A) may be obtained by twisting the multiplication of D(A) by a certain left 2-cocycle which in turn is obtained from the R-matrix of D(A). It was also obtained in ...

let r be a commutative ring with identity and m be a unitary r-module. let : s(m) −! s(m) [ {;} be a function, where s(m) is the set of submodules ofm. suppose n  2 is a positive integer. a proper submodule p of m is called(n − 1, n) − -prime, if whenever a1, . . . , an−1 2 r and x 2 m and a1 . . . an−1x 2p(p), then there exists i 2 {1, . . . , n − 1} such that a1 . . . ai−1ai+1 . . . an−1x...