THE PURE PART OF THE IDEALS IN C(X) Emad Abu Osba

Let C(X) be the ring of all continuous real valued functions dened on a completely regular T1space. For each ideal I in C(X) let mI be the pure part of the ideal I: In this article we show that mI = O(I); where (I) = f2I cl XZ(f). The pure part of many ideals in C(X) is calculated. We found that mCK(X), the pure part of the ideal of functions with compact support, is nitely generated if and only if X-(CK(X)) is com-pact, mCK(X) is countably generated if and only if X-(CK(X)) is Lin-del}o and mCK(X) is generated by a star nite set if and only if X-(CK(X)) is paracompact. Similar results are obtained for the pure part of the ideal C (X), the ideal of functions with pseudocompact support