Let . Similar to it is observed that the sum of any collection of semiprime (resp. prime) ideals in the ring is either or a semiprime (resp. prime) ideal in . For an ideal in , it is observed that and have the same largest -ideal. If is any topological space, we show that there is a zero-dimensional space such that . Consequently, if has only countable number of components, then for some zero-dimensional space . Spaces X for which is regular (called -spaces) are characterized both algebraically and topo log ically and it is shown that -spaces and -spaces coincide when is zero-dimensional. In contrast to , we observe that enjoys the algebraic properties of regularity, -selfinjectivity and some others, whenever has these properties. Finally an example of a space such that is not isomorphic to any is given.
Cite this article
Mostafa Ghadermazi, O.A.S. Karamzadeh, M. Namdari, On the Functionally Countable Subalgebra of . Rend. Sem. Mat. Univ. Padova 129 (2013), pp. 47–69DOI 10.4171/RSMUP/129-4