On S-torsion exact sequences and Si-projective modules (i = 1,2)

2022 ◽  
Author(s):  
Wei Zhao ◽  
Yongyan Pu ◽  
Mingzhao Chen ◽  
Xuelian Xiao
Keyword(s):  
Commutative Ring ◽  
Closed Subset ◽  
Projective Modules ◽  
Exact Sequences

Let [Formula: see text] be a commutative ring and [Formula: see text] a given multiplicative closed subset of [Formula: see text]. In this paper, we introduce the new concept of [Formula: see text]-torsion exact sequences (respectively, [Formula: see text]-torsion commutative diagrams) as a generalization of exact sequences (respectively, commutative diagrams). As an application, they can be used to characterize two classes of modules that are generalizations of projective modules.

ON GRADED S-PRIME SUBMODULES OF GRADED MODULES OVER GRADED COMMUTATIVE RINGS

2021 ◽  
Vol 10 (11) ◽  
pp. 3479-3489
Author(s):  
K. Al-Zoubi ◽  
M. Al-Azaizeh
Keyword(s):  
Abelian Group ◽  
Commutative Ring ◽  
Closed Subset ◽  
Commutative Rings ◽  
Graded Modules ◽  
Prime Submodules ◽  
Prime Submodule

Let $G$ be an abelian group with identity $e$. Let $R$ be a $G$-graded commutative ring with identity, $M$ a graded $R$-module and $S\subseteq h(R)$ a multiplicatively closed subset of $R$. In this paper, we introduce the concept of graded $S$-prime submodules of graded modules over graded commutative rings. We investigate some properties of this class of graded submodules and their homogeneous components. Let $N$ be a graded submodule of $M$ such that $(N:_{R}M)\cap S=\emptyset $. We say that $N$ is \textit{a graded }$S$\textit{-prime submodule of }$M$ if there exists $s_{g}\in S$ and whenever $a_{h}m_{i}\in N,$ then either $s_{g}a_{h}\in (N:_{R}M)$ or $s_{g}m_{i}\in N$ for each $a_{h}\in h(R) $ and $m_{i}\in h(M).$

scholarly journals Rank Element of a Projective Module

1965 ◽  
Vol 25 ◽  
pp. 113-120 ◽  
Author(s):  
Akira Hattori
Keyword(s):  
Finite Group ◽  
Commutative Ring ◽  
Local Ring ◽  
Projective Module ◽  
Projective Modules ◽  
Local Theorem ◽  
Complete Local Ring ◽  
A Finite Group

In § 1 of this note we first define the trace of an endomorphism of a projective module P over a non-commutative ring A. Then we call the trace of the identity the rank element r(P) of P, which we shall illustrate by several examples. For a projective module P over the groupalgebra of a finite group G, the rank element of P is essentially the character of G in P. In § 2 we prove that under certain assumption two projective modules Pi and P2 over an algebra over a complete local ring o are isomorphic if and only if their rank elements are identical. This is a type of proposition asserting that two representations are equivalent if and only if their characters are identical, and in fact, when A is the groupalgebra, the above theorem may be considered as another formulation of Swan’s local theorem [9]).

scholarly journals Adams operations for projective modules over group rings

1997 ◽  
Vol 122 (1) ◽  
pp. 55-71 ◽  
Author(s):  
BERNHARD KÖCK
Keyword(s):  
Finite Group ◽  
Commutative Ring ◽  
Grothendieck Group ◽  
Base Change ◽  
Group Rings ◽  
Projective Modules ◽  
Power Operations ◽  
Definition Of ◽  
A Finite Group ◽  
Twisted Group Ring

Let R be a commutative ring, Γ a finite group acting on R, and let k∈ℕ be invertible in R. Generalizing a definition of Kervaire, we construct an Adams operation ψk on the Grothendieck group and on the higher K-theory of projective modules over the twisted group ring R#Γ. For this, we generalize Atiyah's cyclic power operations and use shuffle products in higher K-theory. For the Grothendieck group, we show that ψk is multiplicative and that it commutes with base change, with the Cartan homomorphism, and with ψl for any other l which is invertible in R.

PROJECTIVE MODULES AND DIVISOR HOMOMORPHISMS

2003 ◽  
Vol 02 (04) ◽  
pp. 435-449 ◽  
Author(s):  
ALBERTO FACCHINI ◽  
FRANZ HALTER-KOCH
Keyword(s):  
Commutative Ring ◽  
Projective Modules ◽  
Finitely Generated ◽  
Commutative Monoids ◽  
Isomorphism Classes

We study some applications of the theory of commutative monoids to the monoid [Formula: see text] of all isomorphism classes of finitely generated projective right modules over a (not necessarily commutative) ring R.

scholarly journals A note on regular modules

1974 ◽  
Vol 11 (3) ◽  
pp. 359-364 ◽  
Author(s):  
V.S. Ramamurthi
Keyword(s):  
Commutative Ring ◽  
Projective Modules ◽  
Von Neumann ◽  
Von Neumann Regular

Kaplansky's observation, namely, a commutative ring R is (von Neumann) regular if and only if each simple R-module is injective, is generalized to projective modules over a commutative ring.

scholarly journals Clean coalgebras and clean comodules of finitely generated projective modules

2021 ◽  
Vol 31 (2) ◽  
pp. 251-260
Author(s):  
N. P. Puspita ◽  
◽  
I. E. Wijayanti ◽  
B. Surodjo ◽  
◽  
...  
Keyword(s):  
Tensor Product ◽  
Commutative Ring ◽  
Sufficient Conditions ◽  
Projective Modules ◽  
Morita Context ◽  
Finitely Generated ◽  
Module Homomorphism

Let R be a commutative ring with multiplicative identity and P is a finitely generated projective R-module. If P∗ is the set of R-module homomorphism from P to R, then the tensor product P∗⊗RP can be considered as an R-coalgebra. Furthermore, P and P∗ is a comodule over coalgebra P∗⊗RP. Using the Morita context, this paper give sufficient conditions of clean coalgebra P∗⊗RP and clean P∗⊗RP-comodule P and P∗. These sufficient conditions are determined by the conditions of module P and ring R.

scholarly journals On Weakly S-Primary Ideals of Commutative Rings

2021 ◽  
Author(s):  
Ece Yetkin Celikeli ◽  
Hani Khashan
Keyword(s):  
Commutative Ring ◽  
Closed Subset ◽  
Commutative Rings ◽  
Primary Ideal ◽  
Ring Extension ◽  
The Stability ◽  
Primary Ideals

Let R be a commutative ring with identity and S be a multiplicatively closed subset of R. The purpose of this paper is to introduce the concept of weakly S-primary ideals as a new generalization of weakly primary ideals. An ideal I of R disjoint with S is called a weakly S-primary ideal if there exists s∈S such that whenever 0≠ab∈I for a,b∈R, then sa∈√I or sb∈I. The relationships among S-prime, S-primary, weakly S-primary and S-n-ideals are investigated. For an element r in any general ZPI-ring, the (weakly) S_{r}-primary ideals are charctarized where S={1,r,r²,⋯}. Several properties, characterizations and examples concerning weakly S-primary ideals are presented. The stability of this new concept with respect to various ring-theoretic constructions such as the trivial ring extension and the amalgamation of rings along an ideal are studied. Furthermore, weakly S-decomposable ideals and S-weakly Laskerian rings which are generalizations of S-decomposable ideals and S-Laskerian rings are introduced.

Homological characterizations of almost Dedekind domains

2019 ◽  
Vol 19 (07) ◽  
pp. 2050128
Author(s):  
Dechuan Zhou ◽  
Hwankoo Kim ◽  
Fanggui Wang ◽  
Kui Hu
Keyword(s):  
Commutative Ring ◽  
Maximal Ideal ◽  
Projective Modules ◽  
Ideal Formula ◽  
Dedekind Domains

Let [Formula: see text] be a commutative ring. In this paper, a class of almost projective modules is introduced. An [Formula: see text]-module [Formula: see text] is said to be almost projective if [Formula: see text] for any [Formula: see text]-module [Formula: see text], where [Formula: see text] is a maximal ideal of [Formula: see text]. It is shown that an [Formula: see text]-module [Formula: see text] satisfying that [Formula: see text] is free over [Formula: see text] for any maximal ideal [Formula: see text] of [Formula: see text] is exactly almost projective. As applications, we characterize almost Dedekind domains in terms of almost projectivity and certain divisibility, respectively.

scholarly journals Finitely projective modules over a Dedekind domain

1978 ◽  
Vol 26 (3) ◽  
pp. 330-336 ◽  
Author(s):  
V. A. Hiremath
Keyword(s):  
Dedekind Domain ◽  
Projective Modules ◽  
Finitely Generated ◽  
Injective Modules ◽  
Exact Sequences

AbstractAs dual to the notion of “finitely injective modules” introduced and studied by Ramamurth and Rangaswamy (1973), we define a right R-module M to be finitely projective if it is projective. with respect to short exact sequences of right R-modules of the form 0 → A → B → C → 0 with C finitely generated. We have completely characterized finitely projective modules over a Dedekind domain. If R is a Dedekind domain, then an R-module M is finitely projective if and only if its reduced part is torsionless and coseparable.For a Dedekind domain R, finite projectivity, unlike projectivity is not hereditary. But it is proved to be pure hereditary, that is, every pure submodule of a finitely projective R-module is finitely projective.

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