On Strongly ?-Regular Group Rings

Chin, A. Y. M.; Chen, Huey Voon

doi:10.1007/s10012-002-0387-z

Cited by 8 publications

(7 citation statements)

References 1 publication

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“…It was proved in [7,Proposition 4] that if R is a Boolean ring and G is a locally finite group, then RG is clean. This is a consequence of the next result, the proof of which uses an idea in [4].…”

Section: Other Extensionsmentioning

confidence: 69%

Strongly Clean Power Series Rings

Chen¹,

Zhou²

2007

Proceedings of the Edinburgh Mathematical Society

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An element a in a ring R with identity is called strongly clean if it is the sum of an idempotent and a unit that commute. And a ∈ R is called strongly π-regular if both chains aR ⊇ a 2 R ⊇ · · · and Ra ⊇ Ra 2 ⊇ · · · terminate. A ring R is called strongly clean (respectively, strongly π-regular) if every element of R is strongly clean (respectively, strongly π-regular). Strongly π-regular elements of a ring are all strongly clean. Let σ be an endomorphism of R. It is proved that foris strongly clean for all n 1 and that the group ring of a locally finite group over a strongly regular or commutative strongly π-regular ring is strongly clean.

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Section: Other Extensionsmentioning

confidence: 69%

Strongly Clean Power Series Rings

Chen¹,

Zhou²

2007

Proceedings of the Edinburgh Mathematical Society

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“…So idempotents of R coincide with idempotents in RG by [8,Lemma 11], and hence every idempotent of RG is a projection. Since R is a ring with artinian prime factors and G is a locally finite 2-group, RG is a strongly π-regular ring by [10,Theorem 3.3]. In view of Theorem 3.6, RG is strongly π- * -regular.…”

Section: Strongly π- * -Regular Ringsmentioning

confidence: 97%

“…Conversely, R is strongly π-regular by [10,Proposition 3.4]. Note that Id(R) ⊆ Id(RG) and all idempotents of RG are projections.…”

Section: Strongly π- * -Regular Ringsmentioning

confidence: 99%

A Note on $*$-Clean Rings

Cui,

Wang

2015

Preprint

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A * -ring R is called (strongly) * -clean if every element of R is the sum of a projection and a unit (which commute with each other). In this note, some properties of * -clean rings are considered. In particular, a new class of *clean rings which called strongly π- * -regular are introduced. It is shown that R is strongly π- * -regular if and only if R is π-regular and every idempotent of R is a projection if and only if R/J(R) is strongly regular with J(R) nil, and every idempotent of R/J(R) is lifted to a central projection of R. In addition, the stable range conditions of * -clean rings are discussed, and equivalent conditions among * -rings related to * -cleanness are obtained.

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“…Before doing that, we just note that (*) if the group ring RG is periodic, then the former ring R is periodic too as its epimorphic image. But, moreover, being periodic, the group ring RG is known to be strongly π-regular, so we extract that G is a torsion group by employing [17,Proposition 3.4]. In the following, we will prove the statement (*) with a different approach.…”

Section: Lemma 11 ([39 Corollary 2]) a Ring R Is Locally Finite If An...mentioning

confidence: 99%

Rings close to periodic with applications to matrix, endomorphism and group rings

Abyzov¹,

Barati²,

Danchev³

2023

Preprint

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We examine those matrix rings whose entries lie in periodic rings equipped with some additional properties. Specifically, we prove that the famous Diesl's question whether or not R being nil-clean implies that M n (R) is nil-clean for all n ≥ 1 is paralleling to the corresponding implication for (Abelian, local) periodic rings. Besides, we study when the endomorphism ring E(G) of an Abelian group G is periodic. Concretely, we establish that E(G) is periodic exactly when G is finite as well as we find a complete necessary and sufficient condition when the endomorphism ring over an Abelian group is strongly m-nil clean for some natural number m thus refining an "old" result concerning strongly nil-clean endomorphism rings. Responding to a question when a group ring is periodic, we show that if R is a right (resp., left) perfect periodic ring and G is a locally finite group, then the group ring RG is periodic, too. We finally find some criteria under certain conditions when the tensor product of two periodic algebras over a commutative ring is again periodic. In addition, some other sorts of rings very close to periodic rings, namely the so-called weakly periodic rings, are also investigated.

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On Strongly ?-Regular Group Rings

Cited by 8 publications

References 1 publication

Strongly Clean Power Series Rings

Strongly Clean Power Series Rings

A Note on $*$-Clean Rings

Rings close to periodic with applications to matrix, endomorphism and group rings

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