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...

let r be a commutative ring with non-zero identity and m be a unital r-module. then the concept of quasi-secondary submodules of m is introduced and some results concerning this class of submodules is obtained

The Generalized Principal Ideal Theorem is one of the cornerstones of dimension theory for Noetherian rings. For an R-module M, we identify certain submodules of M that play a role analogous to that of prime ideals in the ring R. Using this definition, we extend the Generalized Principal Ideal Theorem to modules.

let ( r,m ) be a noetherian local ring, a an ideal of r and m a finitely generated r- module. we investigate some properties of formal local cohomology modules with respect to a serre subcategory. we provide a common language to indicate some properties of formal local cohomology modules. let ( r,m ) be a noetherian local ring, a an ideal of r and m a finitely generated r- module. we investigat...

an r-module m is called strongly noncosingular if it has no nonzero rad-small (cosingular) homomorphic image in the sense of harada. it is proven that (1) an r-module m is strongly noncosingular if and only if m is coatomic and noncosingular; (2) a right perfect ring r is artinian hereditary serial if and only if the class of injective modules coincides with the class of (strongly) noncosingula...

The aim of this note is to generalize the Nakayama lemma to a class of multiplication modules over commutative rings with identity. In this note, by considering the notion of multiplication modules and the product of submodules of them, we state and prove two versions of Nakayama lemma for such modules. In the first version we give some equivalent conditions for faithful finitely generated mult...

‎let $r$ be a domain with quotiont field $k$‎, ‎and‎ ‎let $n$ be a submodule of an $r$-module $m$‎. ‎we say that $n$ is‎ ‎powerful (strongly primary) if $x,yin k$ and‎ ‎$xymsubseteq n$‎, ‎then $xin r$ or $yin r$ ($xmsubseteq n$‎ ‎or $y^nmsubseteq n$ for some $ngeq1$)‎. ‎we show that a submodule‎ ‎with either of these properties is comparable to every prime‎ ‎submodule of $m$‎, ‎also we show tha...

let $r$ be a commutative ring and let $m$ be an $r$-module. in this article, we introduce the concept of the zariski socles of submodules of $m$ and investigate their properties. also we study modules with noetherian second spectrum and obtain some related results.

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...

the generalized principal ideal theorem is one of the cornerstones of dimension theory for noetherian rings. for an r-module m, we identify certain submodules of m that play a role analogous to that of prime ideals in the ring r. using this definition, we extend the generalized principal ideal theorem to modules.