when cc1(x) is an ideal of cc(x)

Cc1(x) denotes the set of all functions f 2 cc(x) that vanish at infinity, that is, for each n 2 n, the set fx 2 x : jf(x)j ≥ 1=ng is compact. it has been shown that cc1(x) coincides with the intersection of all free ideals in cc∗(x). we characterize the spaces x for which cc1(x) forms an ideal in cc(x). in particular, if x is a locally compact, zero-dimensional hausdorff space, then cc1(x) is an ideal in cc(x) if and only if x is c-pseudocompact. moreover, if there exists a function f 2 cc1(x)ncck(x) with z(f) open, then cc1(x) does not form an ideal in cc(x).