A module is called uniseriat if it has a unique composition series of finite length. A ring (always with 1) is called serial if its right and left free modules are direct sums of uniserial modules. Nakayama, who called these rings generalized uniserial rings, proved [21, Theorem 171 that every finitely generated module over a serial ring is a direct sum of uniserial modules. In section one we g...

All rings are commutative with identity and all modules are unital. The tensor product of projective (resp. flat, multiplication) modules is a projective (resp. flat, multiplication) module but not conversely. In this paper we give some conditions under which the converse is true. We also give necessary and sufficient conditions for the tensor product of faithful multiplication Dedekind (resp. ...

In this paper we provide various properties of Rad-⊕-supplemented modules. In particular, we prove that a projective module M is Rad⊕-supplemented if and only if M is ⊕-supplemented, and then we show that a commutative ring R is an artinian serial ring if and only if every left R-module is Rad-⊕-supplemented. Moreover, every left R-module has the property (P ∗) if and only if R is an artinian s...

The aim of this paper is to investigate generalizations locally artinian supplemented modules in module theory, namely radical and strongly modules. We have obtained elementary features for them. Also, we characterized by left perfect rings. Finally, proved that the reduced part a $R$-module has same property over Dedekind domain $R$.

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...

Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. Suppose that $phi:S(M)rightarrow S(M)cup lbraceemptysetrbrace$ be a function where $S(M)$ is the set of all submodules of $M$. A proper submodule $N$ of $M$ is called an $(n-1, n)$-$phi$-classical prime submodule, if whenever $r_{1},ldots,r_{n-1}in R$ and $min M$ with $r_{1}ldots r_{n-1}min Nsetminusphi(N)$, then $r_{1...

We prove a generalization of the Flat Cover Conjecture by showing for any ring R that (1) each (right R-) module has a Ker Ext(−, C)-cover, for any class of pure-injective modules C, and that (2) each module has a Ker Tor(−,B)-cover, for any class of left R-modules B. For Dedekind domains, we describe Ker Ext(−, C) explicitly for any class of cotorsion modules C; in particular, we prove that (1...

Bhargava defined p-orderings of subsets of Dedekind domains and with them studied polynomials which take integer values on those subsets. In analogy with this construction for subsets of Z(p) and p-local integer-valued polynomials in one variable, we define projective p-orderings of subsets of Z(p). With such a projective p-ordering for Z(p) we construct a basis for the module of homogeneous, p...

An i?-module Mis a generator (of the category of modules) provided every module is a homomorphic image of a suitable direct sum of copies of M. Equivalently, some M has R as a summand. Except in the last section, all rings are assumed to be commutative, Noetherian domains, and modules are usually finitely generated. In this context generators are exactly those modules that have non-zero free su...

This research aims to give the decompositions of a finitely generated module over some special ring, such as principal ideal domain and Dedekind domain. One main problems with theory is analyze objects module. was using literature study on modules topics from scientific journals, especially those related theory. And by selective cases we find pattern build conjecture or hypothesis, deductive pr...