It is well known that every prime ideal minimal over a z-ideal is also a z-ideal. The converse is also well known in C（X）. Thus whenever I is an ideal in C（X）, then √I is a z-ideal if and only if I is, in which c...It is well known that every prime ideal minimal over a z-ideal is also a z-ideal. The converse is also well known in C（X）. Thus whenever I is an ideal in C（X）, then √I is a z-ideal if and only if I is, in which case √I = I. We show the same fact for z^-ideals and then it turns out that the sum of a primary ideal and a z-ideal （z^o-ideal） in C（X） which are not in a chain is a prime z-ideal （z^o-ideal）. We also show that every decomposable z-ideal （z^o-ideal） in C（X） is the intersection of a finite number of prime z-ideals （z^o-ideal）. Some counter-examples in general rings and some characterizations for the largest （smallest） z-ideal and z^o-ideal contained in （containing） an ideal are given.展开更多
基金Institute for Studies in Theoretical Physics and Mathematics(IPM),Tehran
文摘It is well known that every prime ideal minimal over a z-ideal is also a z-ideal. The converse is also well known in C（X）. Thus whenever I is an ideal in C（X）, then √I is a z-ideal if and only if I is, in which case √I = I. We show the same fact for z^-ideals and then it turns out that the sum of a primary ideal and a z-ideal （z^o-ideal） in C（X） which are not in a chain is a prime z-ideal （z^o-ideal）. We also show that every decomposable z-ideal （z^o-ideal） in C（X） is the intersection of a finite number of prime z-ideals （z^o-ideal）. Some counter-examples in general rings and some characterizations for the largest （smallest） z-ideal and z^o-ideal contained in （containing） an ideal are given.
