α-Baer rings and some related concepts via C(X)

Abstract

We call a commutative ring R an FIN-ring (resp., FSA-ring) if for any two nitely generated I, J ⊴ R we have Ann(I)+Ann(J) = Ann(I ∩ J) (resp., there is K ⊴ R such that Ann(I) + Ann(J) = Ann(K)). Moreover, we extend this concepts to αIN-rings and αSA-rings where is a cardinal number. The class of FSA-rings includes the class of all SA-rings (hence all IN-rings) and all PP-rings (hence all Baer-rings). In this paper, after giving some properties of αSA-rings, we prove that a reduced ring R is SA if and only if it is an αIN-ring. Consequently, C(X) is an FSA-ring if and only if C(X) is an FIN-ring and equivalently X is an F-space. Moreover, for a commutative ring R, we have shown that R is a Baer-ring if and only if R is a reduced IN-ring. A topological space X is said to be an αUE-space if the closure of any union with cardinal number less than of clopen subsets is open. Topological properties of αUE-spaces are investigated. Finally, we show that a completely regular Hausdorff space X is an αUE-space if and only if C(X) is an αEGE-ring.

Keywords: FSA-ring, Baer-ring, reduced ring, extremally disconnected space, UE-space, basically disconnected space, F-space