Acta mathematica scientia,Series B ›› 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

 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)

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

CLC Number: 

  • 54C40
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