[1] Ying Zhiling, Chen Jianlong,. Rings satisfying UR-stable range [J]. Journal of Southeast University (English Edition), 2008, 24 (4): 545-547. [doi:10.3969/j.issn.1003-7985.2008.04.030]

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Rings satisfying UR-stable range()

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

Volumn:

24

Issue:

2008 4

Page:

545-547

Research Field:

Mathematics, Physics, Mechanics

Publishing date:

2008-12-30

Info

Title:

Rings satisfying UR-stable range

Author(s):

Ying Zhiling;  Chen Jianlong

Department of Mathematics, Southeast University, Nanjing 210096, China

Keywords:

stable range condition;  UR-ring;  matrix extension

PACS:

O153.3

DOI:

10.3969/j.issn.1003-7985.2008.04.030

Abstract:

A ring R is said to be satisfying P-stable range provided that whenever aR+bR=R, there exists y∈P(R)such that a+by is a unit of R, where P(R)is the subset of R which satisfies the property that up, pu∈P(R)for every unit u of R and p∈P(R).By studying this ring, some known results of rings satisfying unit-1 stable range, (S, 2)-stable range, weakly unit 1-stable range and stable range one are unified.An element of a ring is said to be UR if it is the sum of a unit and a regular element and a ring is said to be satisfying UR-stable range if R has P-stable range and P(R)is the set of all UR-elements of R.Some properties of this ring are studied and it is proven that if R satisfies UR-stable range then so does any n×n matrix ring over R.

References:

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[2] Vaserstein L N.Bass’s first stable range condition [J].J Pure Appl Algebra, 1984, 34:319-330.
[3] Menal P, Moncasi J.K₁ of von Neumann regular rings [J].J Pure Appl Algebra, 1984, 33:295-312.
[4] Chen H Y.On stable range conditions [J].Comm Algebra, 2000, 28(8):3913-3924.
[5] Chen H Y.Units, idempotents, and stable range conditions [J].Comm Algebra, 2001, 29(2):703-717.
[6] Chen H Y.Extensions of unit 1-stable range [J].Comm Algebra, 2004, 32(8):3071-3085.
[7] Chen H Y.Extensions of rings with many units [J].Comm Algebra, 2003, 31(5):2403-2426.
[8] Xiao G S, Tong W T.n-clean rings and weakly unit stable range rings [J].Comm Algebra, 2005, 33(5):1501-1517.
[9] Ying Z L, Chen J L.On UR-rings [J].J Math Research and Exposition, to appear.

Memo

Memo:

Biographies: Ying Zhiling(1982—), male, graduate;Chen Jianlong(corresponding author), male, doctor, professor, jlchen@seu.edu.cn.
Foundation items: The National Natural Science Foundation of China(No.10571026), the Specialized Research Fund for the Doctoral Program of Higher Education(No.20060286006).
Citation: Ying Zhiling, Chen Jianlong.Rings satisfying UR-stable range[J].Journal of Southeast University(English Edition), 2008, 24(4):545-547.

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