‎We say that a module $M$ is a emph{cms-module} if‎, ‎for every cofinite submodule $N$ of $M$‎, ‎there exist submodules $K$ and $K^{'}$ of $M$ such that $K$ is a supplement of $N$‎, ‎and $K$‎, ‎$K^{'}$ are mutual supplements in $M$‎. ‎In this article‎, ‎the various properties of cms-modules are given as a generalization of $oplus$-cofinitely supplemented modules‎. ‎In particular‎, ‎we prove tha...

This article introduces the concept of S-2-absorbing primary submodule as a generalization 2-absorbing submodule. Let S be multiplicatively closed subset ring R and M an R-module. A proper N is said to if (N :R M) ? = there exists fixed element s such that whenever abm for some a,b m M, then either sam or sbm sab ?(N M). We give several examples, properties characterizations related concept. Mo...

Let k[V ] be the space of functions from a finite vector space into the algebraically closure of its field of scalars. This paper describes the lattice of subspaces of k[V ] which are invariant under the affine group AGL(V ). The description provides a simple method for finding the submodule generated by any set of functions given as polynomials in the standard coordinates.

Corollary 1.2 Let G be an almost simple classical group with natural module V of dimension n. If G is linear, assume that G does not contain a graph automorphism. Let 2 ≤ k < n − 1, and let K be the stabilizer of a nondegenerate or totally singular k-space of V . Let P be the stabilizer of a singular 1-space of V . Then the permutation module 1P is a submodule of 1 G K unless one of the followi...

in this paper we introduce the notions of uniformly quasi-primary ideals and uniformly classical quasi-primary submodules that generalize the concepts of uniformly primary ideals and uniformly classical primary submodules; respectively. several characterizations of classical quasi-primary and uniformly classical quasi-primary submodules are given. then we investigate for a ring $r$, when any fi...

the submodules with the property of the title ( a submodule $n$ of an $r$-module $m$ is called strongly dense in $m$, denoted by $nleq_{sd}m$, if for any index set $i$, $prod _{i}nleq_{d}prod _{i}m$) are introduced and fully investigated. it is shown that for each submodule $n$ of $m$ there exists the smallest subset $d&apos;subseteq m$ such that $n+d&apos;$ is a strongly dense submodule of $m$...

If N is a submodule of the R-module M , and a ∈ R, let λa : M/N → M/N be multiplication by a. We say that N is a primary submodule of M if N is proper and for every a, λa is either injective or nilpotent. Injectivity means that for all x ∈ M , we have ax ∈ N ⇒ x ∈ N . Nilpotence means that for some positive integer n, aM ⊆ N , that is, a belongs to the annihilator of M/N , denoted by ann(M/N). ...

In this paper, we investigate the ideal theory in BCK-algebras and we define the notions of primary and P -primary ideals. Then we show that in bounded implicative BCK-algebras, if an ideal has a primary decomposition, then it has a reduced primary decomposition. In the follow, we extend the above concepts to X-modules (Extended BCK-modules) and we introduce the notions of primary and P -primar...

Let M be a Hilbert module of holomorphic functions over a natural function algebra A (Ω), where Ω ⊆Cm is a bounded domain. Let M0 ⊆M be the submodule of functions vanishing to order k on a hypersurface Z ⊆ Ω. We describe a method, which in principle may be used, to construct a set of complete unitary invariants for quotient modules Q = M ⊖M0. The invariants are given explicitly in the particula...

Let An(k) be the Weyl algebra, with k a field of characteristic zero. It is known that every projective finitely generated left module is free or isomorphic to a left ideal. Let M be a left submodule of a free module. In this paper we give an algorithm to compute the projective dimension of M . If M is projective and rank(M) ≥ 2 we give a procedure to find a basis. © 2003 Elsevier Ltd. All righ...