On strongly πregular rings and homomorphisms into them

Burgess, W. D.; Menal, Pere

doi:10.1080/00927879808823655

Cited by 48 publications

(24 citation statements)

References 17 publications

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“…A great deal is known about strongly π-regular rings and exchange rings; see for example [4], [15], [16], [20] and [1].…”

Section: Introductionmentioning

confidence: 99%

Strongly 𝜋-regular rings have stable range one

Ara¹

1996

Proc. Amer. Math. Soc.

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Abstract. A ring R is said to be strongly π-regular if for every a ∈ R there exist a positive integer n and b ∈ R such that a n = a n+1 b. For example, all algebraic algebras over a field are strongly π-regular. We prove that every strongly π-regular ring has stable range one. The stable range one condition is especially interesting because of Evans' Theorem, which states that a module M cancels from direct sums whenever End R (M ) has stable range one. As a consequence of our main result and Evans' Theorem, modules satisfying Fitting's Lemma cancel from direct sums.

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“…A great deal is known about strongly π-regular rings and exchange rings; see for example [4], [15], [16], [20] and [1].…”

Section: Introductionmentioning

confidence: 99%

Strongly 𝜋-regular rings have stable range one

Ara¹

1996

Proc. Amer. Math. Soc.

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“…Clearly, local rings are strongly clean. By a result of Burgess and Menal [5], every strongly π -regular ring is strongly clean. In [13], Nicholson gave a direct proof of the result that every strongly π -regular element of a ring is strongly clean, and furthermore he offered the interesting viewpoint that strongly clean elements are natural generalizations of the strongly π -regular elements by establishing the following results: for ϕ ∈ End(M R ), ϕ is strongly π -regular iff there exists a decomposition M = P ⊕Q such that ϕ : P → P is an isomorphism and ϕ : Q → Q is nilpotent; and ϕ is strongly clean iff there exists a decomposition M = P ⊕ Q such that ϕ : P → P and 1 − ϕ : Q → Q are isomorphisms.…”

Section: Introductionmentioning

confidence: 95%

Strong cleanness of the 2×2 matrix ring over a general local ring

Yang

Zhou

2008

Journal of Algebra

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A ring R is called strongly clean if every element of R is the sum of a unit and an idempotent that commute with each other. A recent result of Borooah, Diesl and Dorsey [G. Borooah, A.J. Diesl, T.J. Dorsey, Strongly clean matrix rings over commutative local rings, J. Pure Appl. Algebra 212 (1) (2008) 281-296] completely characterized the commutative local rings R for which M n (R) is strongly clean. For a general local ring R and n > 1, however, it is unknown when the matrix ring M n (R) is strongly clean. Here we completely characterize the local rings R for which M 2 (R) is strongly clean.

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“…Local rings are obviously strongly clean. By Burgess and Menal [1], every strongly π -regular ring is strongly clean, where a ring R is strongly π -regular if the chain aR ⊇ a 2 R ⊇ · · · terminates for every a ∈ R (or equivalently, the chain Ra ⊇ Ra 2 ⊇ · · · terminates for every a ∈ R by Dischinger [5]). In particular, all onesided perfect rings are strongly clean.…”

Section: Introductionmentioning

confidence: 99%

When is the 2×2 matrix ring over a commutative local ring strongly clean?

2006

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A ring R with identity is called strongly clean if every element of R is the sum of an idempotent and a unit that commute. Local rings are strongly clean. It is unknown when a matrix ring is strongly clean. However it is known from [J. Chen, X. Yang, Y. Zhou, On strongly clean matrix and triangular matrix rings, preprint, 2005] that for any prime number p, the 2 × 2 matrix ring M 2 ( Z p ) is strongly clean where Z p is the ring of p-adic integers, but M 2 (Z (p) ) is not strongly clean where Z (p) is the localization of Z at the prime ideal generated by p. Let R be a commutative local ring. A criterion in terms of solvability of a simple quadratic equation in R is obtained for M 2 (R) to be strongly clean. As consequences, M 2 (R) is strongly clean iff M 2 (RJxK) is strongly clean iff M 2 (R[x]/(x n )) is strongly clean iff M 2 (RC 2 ) is strongly clean.

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On strongly πregular rings and homomorphisms into them

Cited by 48 publications

References 17 publications

Strongly 𝜋-regular rings have stable range one

Strongly 𝜋-regular rings have stable range one

Strong cleanness of the 2×2 matrix ring over a general local ring

When is the 2×2 matrix ring over a commutative local ring strongly clean?

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