[1] Wang Zhou, Chen Jianlong,. Extensions of strongly π-regular general rings [J]. Journal of Southeast University (English Edition), 2007, 23 (2): 309-312. [doi:10.3969/j.issn.1003-7985.2007.02.031]

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Extensions of strongly π-regular general rings()

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Journal of Southeast University (English Edition)[ISSN:1003-7985/CN:32-1325/N]

Volumn:

23

Issue:

2007 2

Page:

309-312

Research Field:

Mathematics, Physics, Mechanics

Publishing date:

2007-06-30

Info

Title:

Extensions of strongly π-regular general rings

Author(s):

Wang Zhou;  Chen Jianlong

Department of Mathematics, Southeast University, Nanjing 210096, China

Keywords:

strongly π-regular general ring;  strongly clean general ring;  upper triangular matrix general ring;  trivial extension

PACS:

O153.3

DOI:

10.3969/j.issn.1003-7985.2007.02.031

Abstract:

The concept of the strongly π-regular general ring(with or without unity)is introduced and some extensions of strongly π-regular general rings are considered.Two equivalent characterizations on strongly π-regular general rings are provided.It is shown that I is strongly π-regular if and only if, for each x∈I, xⁿ=x^n+1y=zxn+1 11for n≥1 and y, z∈I if and only if every element of I is strongly π-regular.It is also proved that every upper triangular matrix general ring over a strongly π-regular general ring is strongly π-regular and the trivial extension of the strongly π-regular general ring is strongly clean.

References:

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[4] Chen H Y, Chen M S.On strongly π-regular ideals [J].J Pure Appl Algebra, 2005, 195(1):21-32.
[5] Nicholson W K, Zhou Y Q.Clean general rings [J].J Algebra, 2005, 291(1):279-311.
[6] Ara P.Strongly π-regular rings have stable range one [J].Proc Amer Math Soc, 1996, 124(11):3293-3298.
[7] Anderson D D, Camillo V P.Commutative rings whose elements are a sum of a unit and an idempotent [J].Comm Algebra, 2002, 30(7):3327-3336.
[8] Chen J L, Zhou Y Q.Strongly clean power series rings [J].Proc Edinburgh Math Soc, 2007, 50(1):73-85.
[9] Nicholson W K. Lifting idempotents and exchange rings [J].Trans Amer Math Soc, 1977, 229:269-278.
[10] Nicholson W K, Zhou Y Q.Rings in which elements are uniquely the sum of an idempotent and a unit [J].Glasgow Math J, 2004, 46(2):227-236.

Memo

Memo:

Biographies: Wang Zhou(1979—), male, graduate;Chen Jianlong(corresponding author), male, doctor, professor, jlchen@seu.edu.cn.

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