Displaying similar documents to “Strong separativity over exchange rings”

Diagonal reductions of matrices over exchange ideals

Huanyin Chen (2006)

Czechoslovak Mathematical Journal

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In this paper, we introduce related comparability for exchange ideals. Let I be an exchange ideal of a ring R . If I satisfies related comparability, then for any regular matrix A ∈ M n ( I ) , there exist left invertible U 1 , U 2 ∈ M n ( R ) and right invertible V 1 , V 2 ∈ M n ( R ) such that U 1 V 1 A U 2 V 2 = diag ( e 1 , ⋯ , e n ) for idempotents e 1 , ⋯ , e n ∈ I .

Matrix rings with summand intersection property

F. Karabacak, Adnan Tercan (2003)

Czechoslovak Mathematical Journal

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A ring R has right SIP (SSP) if the intersection (sum) of two direct summands of R is also a direct summand. We show that the right SIP (SSP) is the Morita invariant property. We also prove that the trivial extension of R by M has SIP if and only if R has SIP and ( 1 - e ) M e = 0 for every idempotent e in R . Moreover, we give necessary and sufficient conditions for the generalized upper triangular matrix rings to have SIP.

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...

Nil-clean and unit-regular elements in certain subrings of 𝕄 2 ( ℤ )

Yansheng Wu, Gaohua Tang, Guixin Deng, Yiqiang Zhou (2019)

Czechoslovak Mathematical Journal

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An element in a ring is clean (or, unit-regular) if it is the sum (or, the product) of an idempotent and a unit, and is nil-clean if it is the sum of an idempotent and a nilpotent. Firstly, we show that Jacobson’s lemma does not hold for nil-clean elements in a ring, answering a question posed by Koşan, Wang and Zhou (2016). Secondly, we present new counter-examples to Diesl’s question whether a nil-clean element is clean in a ring. Lastly, we give new examples of unit-regular elements...