Let be a Tychonoff space and be an intermediate subalgebra of , i.e., . We show that such subrings are precisely absolutely convex subalgebras of . An ideal in is said to be a -ideal if , and imply that . We observe that the coincidence of -ideals and -ideals of is equivalent to the equality . This shows that every -ideal in need not be 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 -ideals in is indeed a -ideal in . Next, by studying the sum of -ideals in subrings of the form of , where is an ideal in , we investigate a wide class of examples of subrings of in which the sum of -ideals need not be a -ideal. It is observed that, for every ideal in , the sum of any two -ideals in is a -ideal in or all of if and only if is an -space. This result answers a question raised by Azarpanah, Namdari and Olfati in J. Commut. Algebra 11:4 (2019), 479–509.
"On the sum of $z$-ideals in subrings of $C(X)$." J. Commut. Algebra 12 (4) 459 - 466, Winter 2020. https://doi.org/10.1216/jca.2020.12.459