THE PURE PART OF THE IDEALS IN C(X) Emad Abu Osba University of Petra, Amman 961343, Jordan. emad@uop.edu.jo. H. Al-Ezeh University of Jordan, Amman 11942, Jordan. June 20, 2004 Abstract Let C(X) be the ring of all continuous real valued functions defined on a completely regular T1-space. 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 θ(I) , where θ(I )= f ∈I cl βX Z(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 finitely generated if and only if βX-θ(CK(X)) is com- pact, mCK(X) is countably generated if and only if βX-θ(CK(X)) is Lin- del˝ off and mCK(X) is generated by a star finite set if and only if βX- θ(CK(X)) is paracompact. Similar results are obtained for the pure part of the ideal CΨ(X), the ideal of functions with pseudocompact support. 1. INTRODUCTION Let X be a completely regular T 1 -space, βX the Stone- ∨ Cech compactification of X and υX the Hewitt realcompactification of X. Let C(X) be the ring of all continuous real valued functions defined on X. For each f ∈ C(X), let Z(f ) = {x∈X: f (x) = 0}, coz f = X-Z(f ), the support of f =S X (f ) = cl X coz(f ), S υX (f υ ) = cl υX (υX-Z(f υ )), where f υ is the extension of f to υX, S βX (f β )= cl βX (βX-Z(f β )), where f β is the continuous extension to βX for the function f * (x) =    1 f (x) ≥ 1 f (x) −1 ≤ f (x) ≤ 1 −1 f (x) ≤−1 If I is an ideal in C(X), then coz I =  f ∈I coz f. For each subset A⊆ βX, let M A = {f ∈C(X): A⊆cl βX Z(f )} and O A = {f ∈C(X): A⊆Int βX cl βX Z(f )} = {f ∈C(X): A⊆Int βX Z(f β )}. 1