University of Tehran Journal of Sciences, Islamic Republic of Iran 1016-1104 10 3 1999 09 01 ON COMMUTATIVE GELFAND RINGS - 31435 EN Journal Article 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 https://jsciences.ut.ac.ir/article_31435_7f982c526e15dfa8be4c3eaa864c56ee.pdf