ON BOURBAKI ASSOCIATED PRIME DIVISORS OF AN IDEAL

摘要

Suppose R is a reduced ring. The set of Bourbaki associated prime divisors of an ideal I of R is denoted by (I) and (R) is used instead of (0). Inspired by the concept of fixed-place ideal (fixed-place family), we define the concept of strong fixed-place ideal (strong fixed-place family) and using this concept, we conclude some new results. We show that if I and J are two strong fixed-place ideals of a ring R and I + J = R, then I boolean AND J is a strong fixed-place ideal. Also, we show that the zero ideal of R is strong fixed-place if and only if the zero ideal of R is a fixed- place ideal and R is an i.a.c. ring; if and only if for any subfamily of (R) there is some a in R such that Ann(a) = boolean AND . We prove that B(C(X)) is strong fixed-place if and only if I(X) is a z-embedded subset of X. We deduce that if the zero ideal of R is a strong fixed-place ideal, then there is some extremally disconnected compact space X such that Min(R) Min(C(X)). We prove that if the zero ideal of R is a fixed-place ideal (resp., strong fixed-place ideal), then (resp., ). One of the main questions in algebra is "how can we express the prime ideals of 220f;lambda epsilon?R-lambda by the prime ideals of R-lambda's?". We prove that the zero ideal of R = 220f;lambda epsilon?R-lambda is a fixed-place (strong fixed-place) ideal if and only if the zero ideal of R-lambda is a fixed-place (strong fixed-place) ideal, for every lambda is an element of?; using this result we partially answer to the above question. We conclude that if {D-lambda}lambda is an element of?is an infinite family of integral domains, then and we show that if X is an almost discrete space and I(X) is countable, then |Min(C(X))| = 2 (c).