α-BAER RINGS AND SOME RELATED CON CEPTS VIA C(X)

We call a commutative ring R an FIN-ring (resp., ESA-ring) if for any two finitely generated I, J (sic) JR we have Ann(I) Ann(J) = Arta(J boolean AND J) (resp., there is K (sic) R such that Arta (I) + Ann(J) = APP (K)). Moreover, we extend this concepts to o.IN-rings and aSA-rings where a is a cardinal number. The class of FS.4-rings includes the class of all.9A-rings (hence all IN-rings) and all PP-rings (hence all Baer-rings). In this paper, after giving some properties of alpha SA-rings, We prove that a reduced ring R. is alpha SA if and only if it is an alpha IN-ring. Consequently, C(X) is an ESA-ring if and only if C(X) is an FIN-ring and equivalently X is an F-space. -Moreover, for a commutative ring Ti, we have shown that TI is a Baer-ring if and only if R is a reduced IN-ring. A topological space X is said to be an alpha UE-space if the closure of any union with cardinal number less than cv of clopen subsets is open. Topological properties of alpha UE-spaces are investigated. Finally, we show that a completely regular Hausdorff space X is an alpha UE-space if and only if C(X) is an alpha EGE-ring.