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 2 P...

A submodule [Formula: see text] of is summand absorbing, if implies for any text]. Such submodules often appear in modules over (additively) idempotent semirings, particularly tropical algebra. This paper studies amalgamation and extensions these submodules, more generally upper bound modules.

Let $ m , n in mathbb{N}$, $D$ be a division ring, and $M_{m times n}(D)$ denote the bimodule of all $m times n$ matrices with entries from $D$. First, we characterize one-sided submodules of $M_{m times n}(D)$ in terms of left row reduced echelon or right column reduced echelon matrices with entries from $D$. Next, we introduce the notion of a nest module of matrices with entries from $D$. We ...

Let $G$ be an abelian group with identity $e$. $R$ a $G$-graded commutative ring and $M$ graded $R$-module. In this paper we will obtain some results concerning the generalized 2-absorbing submodules their homogeneous components. Special attention has been paid, when rings are gr-Noetherian, to find extra properties of these submodules.

If R is a commutative ring, then we prove that every finitely generated R-module has a pure-composition series with indecomposable cyclic factors and any two such series are isomorphic if and only if R is a Bézout ring and a CF-ring. When R is a such ring, the length of a pure-composition series of a finitely generated R-module M is compared with its Goldie dimension and we prove that these num...

Let $R$ be a commutative ring with identity and let $M$ be an $R$-module. A proper submodule $P$ of $M$ is called strongly prime submodule if $(P + Rx : M)ysubseteq P$ for $x, yin M$, implies that $xin P$ or $yin P$. In this paper, we study more properties of strongly prime submodules. It is shown that a finitely generated $R$-module $M$ is Artinian if and only if $M$ is Noetherian and every st...

This paper deals with the problem of characterizing submodules of C(T) over certain subalgebras of the disk algebra A. We obtain results that are analogues of the classical characterizations of subspaces of C(T) that are invariant under multiplication by z, i.e., that are submodules over A. These characterizations yield generalizations of Wermer’s maximality theorem applicable to these subalgeb...

The exterior algebra over the centre of a Lie algebra acts on the cohomology of the Lie algebra in a natural way. Focusing on nilpotent Lie algebras, we explore the module structure afforded by this action. We show that for all two-step nilpotent Lie algebras, this module structure is non-trivial, which partially answers a conjecture of Cairns and Jessup [4]. The presence of free submodules ind...

Quasi-cyclic (QC) codes form an important generalization of cyclic codes. It is well know that QC codes of length sl with index s over the finite field F are F[y]-submodules of the ring F[x, y]/< x − 1, y − 1 >. The aim of the present paper, is to study QC codes of length sl with index s over the finite field F and find generator polynomials and generator matrix for these codes. To achieve this...

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