ON THE SUM OF z-IDEALS IN SUBRINGS OF C(X)

Let X be a Tychonoff space and A(X) be an intermediate subalgebra of C(X), i.e., C*(X) subset of A(X) subset of C(X). We show that such subrings are precisely absolutely convex subalgebras of C(X). An ideal I in A(X) is said to be a z(A)-ideal if Z(f) subset of Z(g), f is an element of I and g is an element of A(X) imply that g is an element of I. We observe that the coincidence of z(A)-ideals and z-ideals of A(X) is equivalent to the equality A(X) = C(X). This shows that every z-ideal in A(X) need not be a z(A)-ideal and this is a point which is not considered by D. Rudd in Theorem 4.1 of Michigan Math. J. 17 (1970), 139-141, or by G. Mason in Theorem 3.3 and Proposition 3.5 of Canad. Math. Bull. 23:4 (1980), 437-443. We rectify the induced misconceptions by showing that the sum of z-ideals in A(X) is indeed a z-ideal in A(X). Next, by studying the sum of z-ideals in subrings of the form I + R of C(X), where I is an ideal in C(X), we investigate a wide class of examples of subrings of C(X) in which the sum of z-ideals need not be a z-ideal. It is observed that, for every ideal I in C(X), the sum of any two z-ideals in I + R is a z-ideal in I + R or all of I + R if and only if X is an F-space. This result answers a question raised by Azarpanah, Namdari and Olfati in J. Commut. Algebra 11:4 (2019), 479-509.