On Bourbaki associated prime divisors of an ideal

Abstract

Suppose R is a reduced ring. The set of Bourbaki associated prime divisors of an ideal I of R is denoted by B(I) and B(R) is used instead of B(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 ∩ 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 S of B(R) there is some a in R such that Ann(a) = ∩S. 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 |Min(R)| ≤ 2^{2 |B(R)|} (resp., |Min(R)| = 2^2|B(R)|). One of the main questions in algebra is how can we express the prime ideals of ∏_λ∈Λ R_λ by the prime ideals of R_λ's?". We prove that the zero ideal of R = ∏_λ∈Λ R_λ is a fixed-place (strong fixed-place) ideal if and only if the zero ideal of R_λ is a fixed-place (strong fixed-place) ideal, for every λ∈Λ; using this result we partially answer to the above question. We conclude that if {D_λ}_λ∈Λ is an infinite family of integral domains, then |Min(∏_λ∈Λ D_λ)| = 2^{2 |Λ|} and we show that if X is an almost discrete space and I(X) is countable, then |Min(C(X))| = 2^c.

Mathematics Subject Classification (2010): Primary 13Axx; Secondary 54C40.

Keywords: Minimal prime ideal, fixed-place, strong fixed-place, annihilator condition, rings of continuous functions, filter of ideals