Displaying similar documents to “Associated graded rings and connected sums”

When does the F -signature exist?

Ian M. Aberbach, Florian Enescu (2006)

Annales de la faculté des sciences de Toulouse Mathématiques

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We show that the F -signature of an F -finite local ring R of characteristic p > 0 exists when R is either the localization of an N -graded ring at its irrelevant ideal or Q -Gorenstein on its punctured spectrum. This extends results by Huneke, Leuschke, Yao and Singh and proves the existence of the F -signature in the cases where weak F -regularity is known to be equivalent to strong F -regularity.

On a theorem of McCoy

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

Mathematica Bohemica

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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

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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 commutativity theorem for associative rings

Mohammad Ashraf (1995)

Archivum Mathematicum

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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 ).

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

Tsiu-Kwen Lee, Yiqiang Zhou (2008)

Colloquium Mathematicae

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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

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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...