On commutative Gelfand rings

Abstract

By studying and using the quasi-pure part concept, we improve some statements and show that some assumptions in some articles are superfluous. We give some characterizations of Gelfand rings. For example: we prove that R is Gelfand if and only if m(sum_{α \in A} I_α) = sum_{α \in A} m(I_α), for each family {I_α}_{α \in A} of ideals of R, in addition if R is semiprimitive and Max(R) ⊆ Y ⊆ Spec(R), we show that R is a Gelfand ring if and only if Y is normal. We prove that if R is reduced ring, then R is a von Neumann regular ring if and only if Spec(R) is regular. It has been shown that if R is a Gelfand ring, then Max(R) is a quotient of Spec(R), and sometimes hM(a)’s behave like the zerosets of the space of maximal ideal. Finally, it has been proven that ZMax(C(X)) = {h_M(f) : f \in C(X) } if and only if { h_M(f) : f \in C(X)} is closed under countable intersection if and only if X is pseudocompact.