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 $G$ be a group with identity $e$, $R$ commutative $G$-graded ring unity $1$ and $M$ unital $R$-module. In this article, we introduce the concept of graded $1$-absorbing prime submodule. A proper $R$-submodule $N$ is said to if for all non-unit homogeneous elements $x, y$ element $m$ $xym\in N$, either $xy\in (N :_{R} M)$ or $m\in N$. We show that new generalization submodules at same time i...

Let R be a commutative ring with identity and M be a unital R-module. Then M is called a multiplication module provided for every submodule N of M there exists an ideal I of R such that N = IM. Our objective is to investigate properties of prime and semiprime submodules of multiplication modules. Mathematics Subject Classification: 13C05, 13C13

In this paper, we classify the skew cyclic codes over Fp + vF_p + v^2F_p, where p is a prime number and v^3 = v. Each skew cyclic code is a F_p+vF_p+v^2F_p-submodule of the (F_p+vF_p+v^2F_p)[x;alpha], where v^3 = v and alpha(v) = -v. Also, we give an explicit forms for the generator of these codes. Moreover, an algorithm of encoding and decoding for these codes is presented.

The purpose of this paper is to introduce a new class of domains called quasi-prime algebraic domains. We show that this class is a good candidate for the purpose of denotational semantics of programming languages. This is achieved by exhibiting many constructions usually needed for denotational semantics on quasi-prime algebraic domains. In the first part we motivate the notion of quasi-primes...

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...

we state several conditions under which comultiplication and weak comultiplication modulesare cyclic and study strong comultiplication modules and comultiplication rings. in particular,we will show that every faithful weak comultiplication module having a maximal submoduleover a reduced ring with a finite indecomposable decomposition is cyclic. also we show that if m is an strong comultiplicati...

For σ−PWB extensions, we extend to modules the theory of Gröbner bases of left ideals presented in [5]. As an application, if A is a bijective quasi-commutative σ − PWB extension, we compute the module of syzygies of a submodule of the free module A.

By considering the notion of multiplication modules over a commutative ring with identity, first we introduce the notion product of two submodules of such modules. Then we use this notion to characterize the prime submodules of a multiplication module. Finally, we state and prove a version of Nakayama lemma for multiplication modules and find some related basic results. 1. Introduction. Let R b...

Let R be a commutative ring with identity and M an R–module. If M is either locally cyclic projective or faithful multiplication then M is locally either zero or isomorphic to R. We investigate locally cyclic projective modules and the properties they have in common with faithful multiplication modules. Our main tool is the trace ideal. We see that the module structure of a locally cyclic proje...