# On Bourbaki associated prime divisors of an ideal

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

### 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 IJ 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)| ≤ 22 |B(R)| (resp., |Min(R)| = 22|B(R)|). One of the main questions in algebra is how can we express the prime ideals of ∏λ∈Λ Rλ by the prime ideals of Rλ's?". We prove that the zero ideal of R = ∏λ∈Λ Rλ is a fixed-place (strong fixed-place) ideal if and only if the zero ideal of Rλ 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 {Dλ}λ∈Λ is an infinite family of integral domains, then |Min(∏λ∈Λ Dλ)| = 22 |Λ| and we show that if X is an almost discrete space and I(X) is countable, then |Min(C(X))| = 2c.

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

Published
2019-05-09
Issue
Section
Articles

## Journal Identifiers

eISSN: 1727-933X
print ISSN: 1607-3606