M be a module. In this article we study δ–essential submodules as a dual of δ-small submodules of Zhou where δ = {N ∈ σ[M ] : Rej(N,M) = 0} and M = {N ∈ σ[M ] : N ¿ b N}, and also define μ-M–singular modules as modules N ∈ σ[M ] such that N ∼= K/L for some K ∈ σ[M ] and L is μ–essential in K. By M–M–singular modules and δ–M– singular modules a characterization of GCO–modules, and by FC–M–singul...

A new and conceptually simple procedure is derived for the computation of the maximal reachability submodule of a given submodule of the state space of a linear discrete time system over a Noethenian ring R. The procedure is effective if R is effective and if kernels and intersections can be computed. The procedure is compared with a rather different procedure by Assan e.a. published recently.

Let $R$ be a commutative ring with identity, $S$ multiplicatively closed subset of $R$, and $M$ an $R$-module.
 In this paper, we study investigate some properties $S$-primary submodules $M$. Among the other results, it is shown that this
 class modules contains family primary (resp. $S$-prime) properly.

Our aim in this work is to investigate prime submodules and prove some properties of them. We study the relations between a given module extension submodules. The two modules direct product their quotient are studied investigated.

We show that cohopfian modules with finite exchange have countable exchange. In particular, a module whose endomorphism ring is Dedekind-finite and π-regular has the countable exchange property. We also show that a module whose en-domorphism ring is Dedekind-finite and regular has full exchange. Finally, working modulo the Jacobson radical, we prove that any module with the (C 2) property and a...

Abstract : In this paper we study the properties of the finite topology on the dual of a module over an arbitrary ring. We aim to give conditions when certain properties of the field case can still be found here. Investigating the correspondence between the closed submodules of the dual M∗ of a module M and the submodules of M , we prove some characterizations of PF rings: the up stated corresp...