On Primary Ideals in C(x)
نویسنده
چکیده
2. Primary ideals in C(X). C(X) denotes the ring of real-valued continuous functions on the topological space X, under ordinary pointwise addition and multiplication. By a primary ideal we mean an ideal which is contained in at most one maximal ideal. An O-ideal is an ideal / such that for each pair/i,/2G/ there exists eEI, depending on/i and f2, such that/<«=/.-, for i = l, 2 [7; 4]. A prime-like ideal of C(X) is an ideal P such that if (/, k, e) is any triple of elements of C(X) which satisfies fEP, fe=f and keEP, then kEP [4]. If JEC(X) is an ideal, we let L(J) denote the set {/G C(X):/e =/ ior some eEJ}Obviously L(J)EJWe shall use these additional results from [4]: (i) L(J) is an O-ideal which is a maximal O-ideal if / is a primelike ideal. (ii) If N is an O-ideal in J then NEL(J), so if AT is a maximal 0ideal then N = L(J). (iii) Every prime ideal is a prime-like ideal. The following lemma is used in the proof of Theorem 1 and incidentally to yield an alternate proof of a result in [2].
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