On strongly Π-regular ideals

Chen, Huanyin; Chen, Miaosen

doi:10.1016/j.jpaa.2004.05.006

Cited by 6 publications

(3 citation statements)

References 19 publications

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“…Recall that an element a in a general ring I is called strongly π -regular if there exist n ∈ N and x ∈ I such that a n = a n+1 x and x ∈ comm(a) (see [2,3,17]). The next result is known if a is in a ring R (see [11, Lemma 2.1] and [12,Proposition 4.9]).…”

Section: Theorem 28mentioning

confidence: 99%

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Extensions of quasipolar rings

Gurgun¹

2017

Turk J Math

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An associative ring with identity is called quasipolar provided that for each a ∈ R there exists an idempotent p ∈ R such that p ∈ comm 2 (a) , a + p ∈ U (R) and ap ∈ R qnil . In this article, we introduce the notion of quasipolar general rings (with or without identity). Some properties of quasipolar general rings are investigated. We prove that a general ring I is quasipolar if and only if every element a ∈ I can be written in the form a = s + q where s is strongly regular, s ∈ comm 2 (a) , q is quasinilpotent, and sq = qs = 0 . It is shown that every ideal of a quasipolar general ring is quasipolar. Particularly, we show that R is pseudopolar if and only if R is strongly π -rad clean and quasipolar.

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Section: Theorem 28mentioning

confidence: 99%

“…Ara [1] defined and investigated the notion of an exchange ring without identity. Chen and Chen [3] introduced the concept of strongly π -regular general rings. In [14], Nicholson and Zhou defined the notion of a clean general ring and they extended some of the basic results about clean rings to general rings.…”

Section: Introductionmentioning

confidence: 99%

Extensions of quasipolar rings

Gurgun¹

2017

Turk J Math

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“…This was extended to ideals, i.e., every strongly π-regular ideal of a ring is stable (cf. [6]). The main purpose of this note is to extend these results, and show that every strongly π-regular ideal of a ring is a B-ideal.…”

Section: Introductionmentioning

confidence: 99%

EXTENSIONS OF STRONGLY Π-Regular RINGS

Chen¹,

Kose²,

Kurtulmaz³

2014

Bulletin of the Korean Mathematical Society

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Abstract. An ideal I of a ring R is strongly π-regular if for any x ∈ I there exist n ∈ N and y ∈ I such that x n = x n+1 y. We prove that every strongly π-regular ideal of a ring is a B-ideal. An ideal I is periodic provided that for any x ∈ I there exist two distinct m, n ∈ N such that x m = x n . Furthermore, we prove that an ideal I of a ring R is periodic if and only if I is strongly π-regular and for any u ∈ U (I), u −1 ∈ Z[u].

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Diagonal Reduction Over Sπr Ideals

Chen

2008

Acta Mathematica Scientia

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On strongly Π-regular ideals

Abstract: In this paper, we introduce the concept of strongly -regular ideal of a ring. We prove that every square regular matrix over a strongly -regular ideal of a ring admits a diagonal reduction.

Cited by 6 publications

References 19 publications

Extensions of quasipolar rings

Extensions of quasipolar rings

EXTENSIONS OF STRONGLY Π-Regular RINGS

Diagonal Reduction Over Sπr Ideals

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