Displaying similar documents to “An extension of Zassenhaus' theorem on endomorphism rings”

On a theorem of McCoy

Rajendra K. Sharma, Amit B. Singh (2024)

Mathematica Bohemica

Similarity:

We study McCoy’s theorem to the skew Hurwitz series ring ( HR , ω ) for some different classes of rings such as: semiprime rings, APP rings and skew Hurwitz serieswise quasi-Armendariz rings. Moreover, we establish an equivalence relationship between a right zip ring and its skew Hurwitz series ring in case when a ring R satisfies McCoy’s theorem of skew Hurwitz series.

Rings with divisibility on descending chains of ideals

Oussama Aymane Es Safi, Najib Mahdou, Ünsal Tekir (2024)

Czechoslovak Mathematical Journal

Similarity:

This paper deals with the rings which satisfy D C C d condition. This notion has been introduced recently by R. Dastanpour and A. Ghorbani (2017) as a generalization of Artnian rings. It is of interest to investigate more deeply this class of rings. This study focuses on commutative case. In this vein, we present this work in which we examine the transfer of these rings to the trivial, amalgamation and polynomial ring extensions. We also investigate the relationship between this class of rings...

A note on semilocal group rings

Angelina Y. M. Chin (2002)

Czechoslovak Mathematical Journal

Similarity:

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 commutativity theorem for associative rings

Mohammad Ashraf (1995)

Archivum Mathematicum

Similarity:

Let m > 1 , s ≥ 1 be fixed positive integers, and let R be a ring with unity 1 in which for every x in R there exist integers p = p ( x ) ≥ 0 , q = q ( x ) ≥ 0 , n = n ( x ) ≥ 0 , r = r ( x ) ≥ 0 such that either x p [ x n , y ] x q = x r [ x , y m ] y s or x p [ x n , y ] x q = y s [ x , y m ] x r for all y ∈ R . In the present paper it is shown that R is commutative if it satisfies the property Q ( m ) (i.e. for all x , y ∈ R , m [ x , y ] = 0 implies [ x , y ] = 0 ).

ℵ k -free separable groups with prescribed endomorphism ring

Daniel Herden, Héctor Gabriel Salazar Pedroza (2015)

Fundamenta Mathematicae

Similarity:

We will consider unital rings A with free additive group, and want to construct (in ZFC) for each natural number k a family of ℵ k -free A-modules G which are separable as abelian groups with special decompositions. Recall that an A-module G is ℵ k -free if every subset of size < ℵ k is contained in a free submodule (we will refine this in Definition 3.2); and it is separable as an abelian group if any finite subset of G is contained in a free direct summand of G. Despite the fact that such a...

A unified approach to the Armendariz property of polynomial rings and power series rings

Tsiu-Kwen Lee, Yiqiang Zhou (2008)

Colloquium Mathematicae

Similarity:

A ring R is called Armendariz (resp., Armendariz of power series type) if, whenever ( ∑ i ≥ 0 a i x i ) ( ∑ j ≥ 0 b j x j ) = 0 in R[x] (resp., in R[[x]]), then a i b j = 0 for all i and j. This paper deals with a unified generalization of the two concepts (see Definition 2). Some known results on Armendariz rings are extended to this more general situation and new results are obtained as consequences. For instance, it is proved that a ring R is Armendariz of power series type iff the same is true of R[[x]]. For an injective endomorphism...

Left APP-property of formal power series rings

Zhongkui Liu, Xiao Yan Yang (2008)

Archivum Mathematicum

Similarity:

A ring R is called a left APP-ring if the left annihilator l R ( R a ) is right s -unital as an ideal of R for any element a ∈ R . We consider left APP-property of the skew formal power series ring R [ [ x ; α ] ] where α is a ring automorphism of R . It is shown that if R is a ring satisfying descending chain condition on right annihilators then R [ [ x ; α ] ] is left APP if and only if for any sequence ( b 0 , b 1 , ⋯ ) of elements of R the ideal l R ( ∑ j = 0 ∞ ∑ k = 0 ∞ R α k ( b j ) ) is right s -unital. As an application we give a sufficient condition under which...