On Commutative Gelfand Rings

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(Sigma(alpha is an element of A) I-alpha) = Sigma(alpha is an element of A )m(I-alpha) for each family {I-alpha}alpha is an element of A of ideals of R, in addition if R is semiprimitive and Max(R) subset of Y subset of 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 h(M)(a)'s behave like the zerosets of the space of maximal ideal. Finally, it has been proven that Z(Max(C(X))) = {h(M)(f) : f is an element of C(X)} if and only if {h(M)(f) : f is an element of C(X)} is closed under countable intersection if and only if X is pseudocompact.

Automated summary

By studying and using the quasi-pure part concept, this article improves some statements and provides characterizations of Gelfand rings. Moreover, it explores the relationship between Gelfand rings, their maximal ideals, and specific spaces.