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） S...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.展开更多
基金supported by National Natural Science Foundation of China (10671122)supported by Collegial Natural Science Research Program of Education Department of Jiangsu Province (07KJD110179)
文摘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.
