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