%0 Journal Article %T ON COMMUTATIVE GELFAND RINGS %J Journal of Sciences, Islamic Republic of Iran %I University of Tehran %Z 1016-1104 %D 1999 %\ 09/01/1999 %V 10 %N 3 %P - %! ON COMMUTATIVE GELFAND RINGS %R %X A ring is called a Gelfand ring (pm ring ) if each prime ideal is contained in a unique maximal ideal. For a Gelfand ring R with Jacobson radical zero, we show that the following are equivalent: (1) R is Artinian; (2) R is Noetherian; (3) R has a finite Goldie dimension; (4) Every maximal ideal is generated by an idempotent; (5) Max (R) is finite. We also give the following resu1ts:an ideal of R is uniform, if and only if, it is a minimal ideal; Ass (R) is exactly the set of all maximal ideals which are generated by an idempotent element of R %U https://jsciences.ut.ac.ir/article_31435_ca1164ba1b71c3ad1a40b0d101498d3a.pdf