2-Clean Rings

2009 ◽  
Vol 52 (1) ◽  
pp. 145-153 ◽  
Author(s):  
Z. Wang ◽  
J. L. Chen
Keyword(s):  
Local Ring ◽  
Group Ring ◽  
Endomorphism Ring ◽  
Clean Rings

AbstractA ring R is said to be n-clean if every element can be written as a sum of an idempotent and n units. The class of these rings contains clean rings and n-good rings in which each element is a sum of n units. In this paper, we show that for any ring R, the endomorphism ring of a free R-module of rank at least 2 is 2-clean and that the ring B(R) of all ω × ω row and column-finite matrices over any ring R is 2-clean. Finally, the group ring RCn is considered where R is a local ring.

Some ∗-Clean Group Rings

2015 ◽  
Vol 22 (01) ◽  
pp. 169-180 ◽  
Author(s):  
Yanyan Gao ◽  
Jianlong Chen ◽  
Yuanlin Li
Keyword(s):  
Local Ring ◽  
Group Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Group Rings ◽  
Clean Rings ◽  
Clean Ring ◽  
Commutative Local Ring ◽  
Necessary And Sufficient

A ring with involution ∗ is called ∗-clean if each of its elements is the sum of a unit and a projection. It is obvious that ∗-clean rings are clean. Vaš asked whether there exists a clean ring with involution that is not ∗-clean. In this paper, we investigate when a group ring RG is ∗-clean, where ∗ is the classical involution on RG. We obtain necessary and sufficient conditions for RG to be ∗-clean, where R is a commutative local ring and G is one of the groups C3, C4, S3 and Q8. As a consequence, we provide many examples of group rings which are clean but not ∗-clean.

scholarly journals Commutative feebly nil-clean group rings

2019 ◽  
Vol 11 (2) ◽  
pp. 264-270
Author(s):  
Peter V. Danchev
Keyword(s):  
Abelian Group ◽  
Commutative Ring ◽  
Group Ring ◽  
Group Rings ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Clean Rings ◽  
Unital Ring ◽  
Necessary And Sufficient

Abstract An arbitrary unital ring R is called feebly nil-clean if any its element is of the form q + e − f, where q is a nilpotent and e, f are idempotents with ef = fe. For any commutative ring R and any abelian group G, we find a necessary and sufficient condition when the group ring R(G) is feebly nil-clean only in terms of R, G and their sections. Our result refines establishments due to McGovern et al. in J. Algebra Appl. (2015) on nil-clean rings and Danchev-McGovern in J. Algebra (2015) on weakly nil-clean rings, respectively.

scholarly journals Graded Modules as a Clean Comodule

2020 ◽  
Vol 12 (6) ◽  
pp. 66
Author(s):  
Nikken Prima Puspita ◽  
Indah Emilia Wijayanti ◽  
Budi Surodjo
Keyword(s):  
Group Ring ◽  
Endomorphism Ring ◽  
Module Theory ◽  
Graded Modules ◽  
Clean Ring ◽  
Graded Module

In ring and module theory, the cleanness property is well established. If any element of R can be expressed as the sum of an idempotent and a unit, then R is said to be a clean ring. Moreover, an R-module M is clean if the endomorphism ring of M is clean. We study the cleanness concept of coalgebra and comodules as a dualization of the cleanness in rings and modules. Let C be an R-coalgebra and M be a C-comodule. Since the endomorphism of C-comodule M is a ring, M is called a clean C-comodule if the ring of C-comodule endomorphisms of M is clean. In Brzezi´nski and Wisbauer (2003), the group ring R[G] is an R-coalgebra. Consider M as an R[G]-comodule. In this paper, we have investigated some sucient conditions to make M a clean R[G]-comodule, and have shown that every G-graded module M is a clean R[G]-comodule if M is a clean R-module.

Artin Rings with Two-Generated Ideals

1992 ◽  
Vol 35 (1) ◽  
pp. 133-135 ◽  
Author(s):  
David E. Rush
Keyword(s):  
Local Ring ◽  
Complete Intersection ◽  
Homomorphic Image ◽  
Endomorphism Ring ◽  
Commutative Group ◽  
Group Rings ◽  
One Dimensional ◽  
Intersection Ring

AbstractIt is shown that each commutative Artin local ring having each of its ideals generated by two elements is the homomorphic image of a one-dimensional local complete intersection ring which also has each of its ideals generated by two elements. It is indicated how this can be applied to show that the property that each ideal is projective over its endomorphism ring does not pass to homomorphic images, and in determining the commutative group rings with the two-generator property.

scholarly journals KERNELS OF MORPHISMS BETWEEN INDECOMPOSABLE INJECTIVE MODULES

2010 ◽  
Vol 52 (A) ◽  
pp. 69-82 ◽  
Author(s):  
ALBERTO FACCHINI ◽  
ŞULE ECEVIT ◽  
M. TAMER KOŞAN
Keyword(s):  
Local Ring ◽  
Endomorphism Ring ◽  
Weak Form ◽  
Indecomposable Module ◽  
Endomorphism Rings ◽  
Direct Sums ◽  
Indecomposable Modules ◽  
Direct Sum ◽  
Injective Modules ◽  
Maximal Ideals

AbstractWe show that the endomorphism rings of kernels ker ϕ of non-injective morphisms ϕ between indecomposable injective modules are either local or have two maximal ideals, the module ker ϕ is determined up to isomorphism by two invariants called monogeny class and upper part, and a weak form of the Krull–Schmidt theorem holds for direct sums of these kernels. We prove with an example that our pathological decompositions actually take place. We show that a direct sum ofnkernels of morphisms between injective indecomposable modules can have exactlyn! pairwise non-isomorphic direct-sum decompositions into kernels of morphisms of the same type. IfERis an injective indecomposable module andSis its endomorphism ring, the duality Hom(−,ER) transforms kernels of morphismsER→ERinto cyclically presented left modules over the local ringS, sending the monogeny class into the epigeny class and the upper part into the lower part.

Semigroup identities on units of integral group rings

1997 ◽  
Vol 39 (1) ◽  
pp. 1-6 ◽  
Author(s):  
Michael A. Dokuchaev ◽  
Jairo Z. Gonçalves
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Group Ring ◽  
Torsion Group ◽  
Group Rings ◽  
Integral Group ◽  
Semigroup Identity ◽  
Integral Group Rings ◽  
Group Of Units ◽  
Semigroup Identities

AbstractLet U(RG) be the group of units of a group ring RG over a commutative ring R with 1. We say that a group is an SIT-group if it is an extension of a group which satisfies a semigroup identity by a torsion group. It is a consequence of the main result that if G is torsion and R = Z, then U(RG) is an SIT-group if and only if G is either abelian or a Hamiltonian 2-group. If R is a local ring of characteristic 0 only the first alternative can occur.

scholarly journals The rank of syzygies under the action by a finite group

1978 ◽  
Vol 71 ◽  
pp. 1-12 ◽  
Author(s):  
Shiro Goto
Keyword(s):  
Finite Group ◽  
Local Ring ◽  
Group Ring ◽  
Group Algebra ◽  
Maximal Ideal ◽  
Residue Field ◽  
Noetherian Local Ring ◽  
Trace Ideal ◽  
A Finite Group ◽  
Twisted Group Ring

Let S be a Noetherian local ring with maximal ideal J and k the residue field of S. Let G be a finite group of order n and suppose that G acts on S as automorphisms. Let R = SG and I = JG. We denote by S[G] (resp. R[G]) the twisted group ring of G over S (resp. the group algebra of G over R). Recall that the multiplication of S[G] is defined as follows : sg · th = sg(t) · gh for s, t ∊ S and g, h ∊ G. Let tG(S) = {Σg ∈ Gg(s)/s ∈ S} and call it the trace ideal of S. Note that tG(S) = R if n is a unit of S. We say that S has a normal basis if S ≅ R[G] as R[G]-modules. This condition says that there is an element s of S so that {g(s)}g ∈ G forms an R-free basis of S.

scholarly journals EXISTENCE OF ALMOST SPLIT SEQUENCES VIA REGULAR SEQUENCES

2013 ◽  
Vol 88 (2) ◽  
pp. 218-231 ◽  
Author(s):  
HOSSEIN ESHRAGHI
Keyword(s):  
Local Ring ◽  
Endomorphism Ring ◽  
Krull Dimension ◽  
Inductive Argument ◽  
Split Sequence ◽  
Regular Sequences ◽  
Complete Local Ring ◽  
Almost Split Sequence ◽  
Almost Split Sequences

AbstractLet $(R, \mathfrak{m})$ be a Cohen–Macaulay complete local ring. We will apply an inductive argument to show that for every nonprojective locally projective maximal Cohen–Macaulay object $ \mathcal{X} $ of the morphism category of $R$ with local endomorphism ring, there exists an almost split sequence ending in $ \mathcal{X} $. Regular sequences are exploited to reduce the Krull dimension of $R$ on which the inductive argument is established. Moreover, the Auslander–Reiten translate of certain objects is described.

M-SP-INJECTIVE MODULES

2012 ◽  
Vol 05 (01) ◽  
pp. 1250005
Author(s):  
V. Kumar ◽  
A. J. Gupta ◽  
B. M. Pandeya ◽  
M. K. Patel
Keyword(s):  
Local Ring ◽  
Endomorphism Ring ◽  
Injective Module ◽  
Injective Modules ◽  
Finite Dimensional

In this paper we study M-small principally injective (in short, M-sp-injective) module which is the generalization of M-principally injective module. We prove that if M is finite dimensional and quasi-sp-injective then its endomorphism ring S is semi-local ring. We characterize the M-sp-injective module with the help of epi-retractable modules.

Rings whose Elements are the Sum of a Tripotent and an Element from the Jacobson Radical

2019 ◽  
Vol 62 (4) ◽  
pp. 810-821 ◽  
Author(s):  
M. Tamer Koşan ◽  
Tülay Yildirim ◽  
Y. Zhou
Keyword(s):  
Abelian Group ◽  
Nilpotent Group ◽  
Group Ring ◽  
Jacobson Radical ◽  
Sufficient Conditions ◽  
Morita Context ◽  
Locally Finite ◽  
Boolean Rings ◽  
Clean Rings ◽  
Necessary And Sufficient

AbstractThis paper is about rings $R$ for which every element is a sum of a tripotent and an element from the Jacobson radical $J(R)$. These rings are called semi-tripotent rings. Examples include Boolean rings, strongly nil-clean rings, strongly 2-nil-clean rings, and semi-boolean rings. Here, many characterizations of semi-tripotent rings are obtained. Necessary and sufficient conditions for a Morita context (respectively, for a group ring of an abelian group or a locally finite nilpotent group) to be semi-tripotent are proved.

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