# 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