[1] Huang Qinghe, Chen Jianlong,. Exchange general rings [J]. Journal of Southeast University (English Edition), 2006, 22 (4): 589-592. [doi:10.3969/j.issn.1003-7985.2006.04.029]
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Exchange general rings(
)
Exchange 一般环
)Journal of Southeast University (English Edition)[ISSN:1003-7985/CN:32-1325/N]
- Volumn:
- 22
- Issue:
- 2006 4
- Page:
- 589-592
- Research Field:
- Mathematics, Physics, Mechanics
- Publishing date:
- 2006-12-30
Info
- Title:
- Exchange general rings
- Exchange 一般环
- Author(s):
- Huang Qinghe; Chen Jianlong
- School of Sciences, Southeast University, Nanjing 210096, China
- Keywords:
- exchange ring; clean element; stable range one; ideal extension
- PACS:
- O153
- DOI:
- 10.3969/j.issn.1003-7985.2006.04.029
- Abstract:
- The exchange rings without unity, first introduced by Ara, are further investigated.Some new characterizations and properties of exchange general rings are given.For example, a general ring I is exchange if and only if for any left ideal L of I and (-overa)=(-overa)2∈I/L2, there exists w∈r.ureg(I)such that (-overw)=(-overa);E(R, I)(the ideal extension of a ring R by its ideal I)is an exchange ring if and only if R and I are both exchange.Furthermore, it is presented that if I is a two-sided ideal of a unital ring R and I is an exchange general ring, then every central element of I is a clean element in I.
- 进一步研究了由Ara首次引入并研究的没有单位元的exchange环.给出了它的一些新的等价刻画和性质.例如:一个一般环I 是exchange的当且仅当对它的任意理想L以及(-overa)=(-overa)2∈I/L2, 存在w∈r.ureg(I)使得(-overw)=(-overa); E(R, I)(环R通过它的理想I生成的理想扩张)是一个exchange环当且仅当R和I都是exchange环.还证明了如果环R的双边理想I是一个exchange一般环, 则I的每一个中心元素都是I中一个 clean 元素.
References:
[1] Crawley P, Jónsson B.Refinements for infinite direct decompositions of algebra systems[J].Pacific J Math, 1964, 14(3):797-855.
[2] Warfield R B. Exchange rings and decompositions of modules[J].Math Ann, 1972, 199(1):31-36.
[3] Ara P.Extensions of exchange rings[J].J Algebra, 1997, 197(2):409-423.
[4] Nicholson W K, Zhou Y.Clean general rings[J].J Algebra, 2005, 291(1):297-311.
[5] Zhang Hongbo.On cancellations property[D].Nanjing:Department of Mathematics of Nanjing University, 2004.(in Chinese)
[6] Dinesh Khurana, Lam T Y.Rings with internal cancellation[J]. J Algebra, 2005, 284(1):203-235.
[7] Hong C Y, Kim N K, Lee Y.Exchange rings and their extensions[J].Pure Appl Algebra, 2003, 179(1):117-126.
Memo
- Memo:
- Biographies: Huang Qinghe(1981—), female, graduate;Chen Jianlong(corresponding author), male, professor, jlchen@seu.edu.cn.
Last Update: 2006-12-20