Study of Morita contexts

Tang, Gaohua; Li, Chunna; Zhou, Yiqiang

doi:10.1080/00927872.2012.748327

Cited by 39 publications

(12 citation statements)

References 16 publications

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“…Moreover, A, B are strongly π-regular rings by [10,Corollary 3.7]. With the assumption that MN, NM are nilpotent ideals of A and B respectively, we infer that R is strongly π-regular by [29,Theorem 3.5]. Hence J(R) is nil, and so R is strongly nil-clean by Theorem 2.7.…”

Section: Canonically M/m 0 Is An A/j(a) B/j(b) -Bimodule and N/n 0 mentioning

confidence: 83%

Nil-clean and strongly nil-clean rings

Koşan

Wang

Zhou

2016

Journal of Pure and Applied Algebra

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Section: Canonically M/m 0 Is An A/j(a) B/j(b) -Bimodule and N/n 0 mentioning

confidence: 83%

Nil-clean and strongly nil-clean rings

Koşan

Wang

Zhou

2016

Journal of Pure and Applied Algebra

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“…In [10], generalized matrix ring K s (R) over a ring R is defined and investigated in detail. Addition in K s (R) is componentwise and multiplication is given by…”

Section: Extensionsmentioning

confidence: 99%

Rings having normality in terms of the Jacobson radical

et al. 2018

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A ring R is defined to be J-normal if for any a, r ∈ R and idempotent e ∈ R, ae = 0 implies Rera ⊆ J (R), where J (R) is the Jacobson radical of R. The class of J-normal rings lies between the classes of weakly normal rings and left min-abel rings. It is proved that R is J-normal if and only if for any idempotent e ∈ R and for any r ∈ R, R(1 − e)re ⊆ J (R) if and only if for any n ≥ 1, the n × n upper triangular matrix ring U n (R) is a J-normal ring if and only if the Dorroh extension of R by Z is J-normal. We show that R is strongly regular if and only if R is J-normal and von Neumann regular. For a J-normal ring R, it is obtained that R is clean if and only if R is exchange. We also investigate J-normality of certain subrings of the ring of 2 × 2 matrices over R. Mathematics Subject Classification IntroductionThroughout this work, every ring is associative with identity unless otherwise stated. Recently, some kinds of normality for rings have been investigated in the literature. For instance, the notion of quasi-normality of rings was defined in [13], that is, a ring R is called quasi-normal if ae = 0 implies ea Re = 0 for every nilpotent element a and idempotent e of R. On the other hand, another kind of normality was introduced in [14], namely, a ring R is said to be weakly normal if for all elements a, r and e 2 = e of R, ae = 0 implies that Rera is a nil left ideal of R. It is seen that the notion of a weakly normal ring is a generalization of that of a quasi-normal ring.The Jacobson radical is an important tool for studying the structure of a ring. In the light of aforementioned concepts, it is a reasonable question that what kind of properties does a ring gain when it satisfies normality in terms of its Jacobson radical? This question is one of the motivations to deal with the notion of normality

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“…When s = 1, K 1 (R) is just the matrix ring M 2 (R), but K s (R) can be significantly different from M 2 (R). In fact, for a local ring R and s ∈ C(R), K s (R) ∼ = K 1 (R) iff s ∈ U(R) (see [6,Lemma 3 and Corollary 2] and [12,Corollary 4.10]). The reason for conducting this work is simple: many new families of strongly clean rings can be constructed through K s (R).…”

Section: Introductionmentioning

confidence: 97%

Strong cleanness of generalized matrix rings over a local ring

Tang

Zhou

2012

Linear Algebra and its Applications

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Let K s (R) be the generalized matrix ring over a ring R with multiplier s. For a general local ring R and a central element s in the Jacobson radical of R, necessary and sufficient conditions are obtained for K s (R) to be a strongly clean ring. For a commutative local ring R and an arbitrary element s in R, criteria are obtained for a single element of K s (R) to be strongly clean and, respectively, for the ring K s (R) to be strongly clean. Specializing to s = 1 yields some known results. New families of strongly clean rings are presented.

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Study of Morita contexts

Cited by 39 publications

References 16 publications

Nil-clean and strongly nil-clean rings

Nil-clean and strongly nil-clean rings

Rings having normality in terms of the Jacobson radical

Strong cleanness of generalized matrix rings over a local ring

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