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A note on semilocal group rings

Angelina Y. M. Chin (2002)

Czechoslovak Mathematical Journal

Let R be an associative ring with identity and let J ( R ) denote the Jacobson radical of R . R is said to be semilocal if R / J ( R ) is Artinian. In this paper we give necessary and sufficient conditions for the group ring R G , where G is an abelian group, to be semilocal.

A remark on power series rings.

Paul M. Cohn (1992)

Publicacions Matemàtiques

A trivializability principle for local rings is described which leads to a form of weak algorithm for local semifirs with a finitely generated maximal ideal whose powers meet in zero.

A representation theorem for Chain rings

Yousef Alkhamees, Hanan Alolayan, Surjeet Singh (2003)

Colloquium Mathematicae

A ring A is called a chain ring if it is a local, both sided artinian, principal ideal ring. Let R be a commutative chain ring. Let A be a faithful R-algebra which is a chain ring such that Ā = A/J(A) is a separable field extension of R̅ = R/J(R). It follows from a recent result by Alkhamees and Singh that A has a commutative R-subalgebra R₀ which is a chain ring such that A = R₀ + J(A) and R₀ ∩ J(A) = J(R₀) = J(R)R₀. The structure of A in terms of a skew polynomial ring over R₀ is determined.

Automorphism liftable modules

Chelliah Selvaraj, Sudalaimuthu Santhakumar (2018)

Commentationes Mathematicae Universitatis Carolinae

We introduce the notion of an automorphism liftable module and give a characterization to it. We prove that category equivalence preserves automorphism liftable. Furthermore, we characterize semisimple rings, perfect rings, hereditary rings and quasi-Frobenius rings by properties of automorphism liftable modules. Also, we study automorphism liftable modules with summand sum property (SSP) and summand intersection property (SIP).

Characterizations of semiperfect and perfect rings.

Weimin Xue (1996)

Publicacions Matemàtiques

We characterize semiperfect modules, semiperfect rings, and perfect rings using locally projective covers and generalized locally projective covers, where locally projective modules were introduced by Zimmermann-Huisgen and generalized locally projective covers are adapted from Azumaya’s generalized projective covers.

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