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(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 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 \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.