Acta mathematica scientia, Series B >
TOPOLOGICAL CHARACTERIZATIONS OF THE EXTENDING PROPERTY OF RINGS
Received date: 2007-08-30
Revised date: 2008-06-09
Online 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)
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
LU Dan-Cheng , WU Tong-Suo . TOPOLOGICAL CHARACTERIZATIONS OF THE EXTENDING PROPERTY OF RINGS[J]. Acta mathematica scientia, Series B, 2010 , 30(4) : 1086 -1092 . DOI: 10.1016/S0252-9602(10)60105-1
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