# associated prime ideal

## associated prime ideal

[ə′sō·sē‚ād·əd ′prīm ‚ī·dēl]
(mathematics)
A prime ideal I in a commutative ring R is said to be associated with a module M over R if there exists an element x in M such that I is the annihilator of x.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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In our case, we only have to check finitely many p's, namely, the associated prime ideals of fB.
[intersection] [N.sub.r] a primary decomposition of N in M with associated prime ideals [Mathematical Expression Omitted].
f) Let I [subset] N [(omega)] be an ideal with compact zero-set, and let I = [I.sub.1][subset]...[subset][I.sub.r] be a primary decomposition with associated prime ideals [p.sub.i] = [Mathematical Expressions Omitted], 1 [less than or equal to] i [less than or equal to] r.
If [[intersection].sub.1[less than or equal to]i[less than or equal to]m][Q.sub.i] is an irredundant intersection of primary ideal [Q.sub.i] with associated prime ideals [P.sub.i], 1 [less than or equal to] i [less than or equal to] m, then the closure of I is equal to the intersection of those [Q.sub.i] with [Q.sub.i] + A [not equal to] M.
When R is Noetherian, every prime ideal has finite height and the height of an arbitrary ideal is defined to be the minimum of the heights of its associated prime ideals. The Krull dimension of R is the supremum of the heights of the proper ideals of R.
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