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[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()
具有UR-稳定度的环
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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
具有UR-稳定度的环
Author(s):
Ying Zhiling Chen Jianlong
Department of Mathematics, Southeast University, Nanjing 210096, China
应志领 陈建龙
东南大学数学系, 南京 210096
Keywords:
stable range condition UR-ring matrix extension
稳定度条件 UR-环 矩阵扩张
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, puP(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.
R具有P稳定度是指若有aR+bR=R, 则存在y∈P(R)使得a+byR中的可逆元.其中P(R)是环R的子集并满足如下性质:对于任意的可逆元up∈P(R)都有up, puP(R).通过对环R的研究, 统一了关于具有可逆-1稳定度、(S, 2)-稳定度、弱可逆-1稳定度和稳定度为1的环的一些已知结果.当环的一个元素是一个可逆元和一个正则元之和, 则称这个元素为UR.如果环R具有P 稳定度且P(R)是环中所有UR元素组成的集合, 则称环R具有UR-稳定度.研究了该环的性质, 并证明了如果R具有UR-稳定度, 则R上的任意n阶矩阵环也具有UR-稳定度.

References:

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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.
Last Update: 2008-12-20