TOPOLOGICAL CHARACTERIZATIONS OF THE EXTENDING PROPERTY OF RINGS |
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Authors: | Lu Dancheng Wu Tongsuo |
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Institution: | [1]Department of Mathematics, Soochow University, Suzhou 215006, China [2]Department of Mathematics, Shanghai Jiaotong University, Shanghai 200240, China |
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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. |
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Keywords: | Extending Rings extremely disconnected prime spectrum |
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