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 p α∈a iα
p = α∈a m(iα), for each family {iα}α∈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 z max(c(x))
= {hm(f) : f ∈ c(x)} if and only if {hm(f) : f ∈ c(x)} is closed under countable intersection if and only if x is pseudocompact