数学物理学报(英文版) ›› 2010, Vol. 30 ›› Issue (4): 1086-1092.doi: 10.1016/S0252-9602(10)60105-1

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TOPOLOGICAL CHARACTERIZATIONS OF THE EXTENDING PROPERTY OF RINGS

卢丹诚1|武同锁2   

  1. 1. Department of Mathematics, Soochow University, Suzhou 215006, China； 2. Department of Mathematics, Shanghai Jiaotong University, Shanghai 200240, China
  • 收稿日期:2007-08-30 修回日期:2008-06-09 出版日期:2010-07-20 发布日期:2010-07-20
  • 基金资助:

    This work is supported by National Natural Science Foundation of China (10671122) and partly supported by Collegial Natural Science Research Program of Education Department of Jiangsu Province (07KJD110179)

TOPOLOGICAL CHARACTERIZATIONS OF THE EXTENDING PROPERTY OF RINGS

 LU Dan-Cheng1, WU Tong-Suo2   

  1. 1. Department of Mathematics, Soochow University, Suzhou 215006, China； 2. Department of Mathematics, Shanghai Jiaotong University, Shanghai 200240, China
  • Received:2007-08-30 Revised:2008-06-09 Online:2010-07-20 Published:2010-07-20
  • Supported by:

    This work is supported by National Natural Science Foundation of China (10671122) and partly supported by Collegial Natural Science Research Program of Education Department of Jiangsu Province (07KJD110179)

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摘要：

A commutative ring R is called extending  if every ideal is essential in a direct summand of RR. The following results are proved: (1) C(X) is an extending ring if and only if X is extremely disconnected; (2) Spec(R) is extremely disconnected and R is semiprime if and only if R is a nonsingular extending ring; (3) Spec(R) is extremely disconnected if and only if R/N}(R) is an extending ring, where N(R) consists of all nilpotent elements of R. As an application, it is also shown that any Gelfand nonsingular extending ring is clean.

关键词: Extending Rings, extremely disconnected, prime spectrum

Abstract:

A commutative ring R is called extending  if every ideal is essential in a direct summand of RR. The following results are proved: (1) C(X) is an extending ring if and only if X is extremely disconnected; (2) Spec(R) is extremely disconnected and R is semiprime if and only if R is a nonsingular extending ring; (3) Spec(R) is extremely disconnected if and only if R/N}(R) is an extending ring, where N(R) consists of all nilpotent elements of R. As an application, it is also shown that any Gelfand nonsingular extending ring is clean.

Key words: Extending Rings, extremely disconnected, prime spectrum

中图分类号: 

  • 54C40
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