scholarly journals Group-regular rings

Filomat ◽  
2019 ◽  
Vol 33 (11) ◽  
pp. 3551-3560
Author(s):  
Xavier Mary ◽  
Pedro Patrício
Keyword(s):  
Stable Range ◽  
Stable Range One ◽  
Unital Rings ◽  
Regular Rings

We propose different generalizations of unit-regularity of elements in general rings (non necessarily unital rings). We then study general rings for which all elements have these properties. We notably compare them with unit-regular ideals and general rings with stable range one. We also prove that these rings are morphic rings.

On stable range one property and strongly $$\pi $$-regular rings

2020 ◽  
Vol 31 (5-6) ◽  
pp. 1047-1056
Author(s):  
Rachida El Khalfaoui ◽  
Najib Mahdou
Keyword(s):  
Stable Range ◽  
Stable Range One ◽  
Regular Rings

scholarly journals A note on regular rings with stable range one

2002 ◽  
Vol 31 (7) ◽  
pp. 449-450
Author(s):  
H. V. Chen ◽  
A. Y. M. Chin
Keyword(s):  
Regular Ring ◽  
Elementary Proof ◽  
Stable Range ◽  
Stable Range One ◽  
Regular Rings

It is known that a regular ring has stable range one if and only if it is unit regular. The purpose of this note is to give an independent and more elementary proof of this result.

scholarly journals On strongly pi-regular rings of stable range one

1995 ◽  
Vol 51 (3) ◽  
pp. 433-437 ◽  
Author(s):  
Hua-Ping Yu ◽  
Victor P. Camilo
Keyword(s):  
Regular Element ◽  
Regular Ring ◽  
Associative Ring ◽  
Stable Range ◽  
Von Neumann ◽  
Stable Range One ◽  
Von Neumann Regular ◽  
Open Question ◽  
Regular Rings

An associative ring R is said to have stable range one if for any a, b ∈ R satisfying aR + bR = R, there exists y ∈ R such that a + by is right (equivalently, left) invertible. Call a ring R strongly π-regular if for every element a ∈ R there exist a number n (depending on a) and an element x ∈ R such that an = an+1x. It is an open question whether all strongly π-regular rings have stable range one. The purpose of this note is to prove the following Theorem: If R is a strongly π-regular ring with the property that all powers of every nilpotent von Neumann regular element are von Neumann regular in R, then R has stable range one.

scholarly journals Stable range one for rings with central units

2020 ◽  
Vol 48 (11) ◽  
pp. 4767-4773
Author(s):  
Paula A. A. B. Carvalho ◽  
Christian Lomp ◽  
Jerzy Matczuk
Keyword(s):  
Stable Range ◽  
Stable Range One

scholarly journals Exchange rings having stable range one

2001 ◽  
Vol 25 (12) ◽  
pp. 763-770 ◽  
Author(s):  
Huanyin Chen
Keyword(s):  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Stable Range ◽  
Exchange Ring ◽  
Stable Range One ◽  
Exchange Rings

We investigate the sufficient conditions and the necessary conditions on an exchange ringRunder whichRhas stable range one. These give nontrivial generalizations of Theorem 3 of V. P. Camillo and H.-P. Yu (1995), Theorem 4.19 of K. R. Goodearl (1979, 1991), Theorem 2 of R. E. Hartwig (1982), and Theorem 9 of H.-P. Yu (1995).

scholarly journals On regular rings with stable range 2

1982 ◽  
Vol 24 (1) ◽  
pp. 25-40 ◽  
Author(s):  
Pere Menal ◽  
Jaume Moncasi
Keyword(s):  
Stable Range ◽  
Regular Rings

Exchange rings having ideal-stable range one

2001 ◽  
Vol 44 (5) ◽  
pp. 579-586 ◽  
Author(s):  
Huanyin Chen ◽  
Fu’an Li
Keyword(s):  
Stable Range ◽  
Stable Range One ◽  
Exchange Rings

scholarly journals Invo-regular unital rings

2018 ◽  
Vol 72 (1) ◽  
pp. 45
Author(s):  
Peter V. Danchev
Keyword(s):  
Regular Ring ◽  
A Priori ◽  
Proper Subclass ◽  
In Trans ◽  
Unital Rings ◽  
Regular Rings

It was asked by Nicholson (Comm. Algebra, 1999) whether or not unit-regular rings are themselves strongly clean. Although they are clean as proved by Camillo-Khurana (Comm. Algebra, 2001), recently Nielsen and Ster showed in Trans. Amer. Math. Soc., 2018 that there exists a unit-regular ring which is not strongly clean. However, we define here a proper subclass of rings of the class of unit-regular rings, called invo-regular rings, and establish that they are strongly clean. Interestingly, without any concrete indications a priori, these rings are manifestly even commutative invo-clean as defined by the author in Commun. Korean Math. Soc., 2017.

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