associated prime

Let $R$ be a ring, and let $M$ be an $R$-module. A prime ideal $P$ of $R$ is an for $M$ if $P=\mathrm{ann}(X)$, the annihilator of some nonzero submodule $X$ of $M$.

Note that if this is the case, then the module ${\mathrm{ann}}_M(P)$ contains $X$, has $P$ as its annihilator, and is a faithful (http://planetmath.org/FaithfulModule) $(R/P)$-module.

If, in addition, $P$ is equal to the annihilator of a submodule of $M$ that is a fully faithful (http://planetmath.org/FaithfulModule) $(R/P)$-module, then we call $P$ an of $M$.

Title	associated prime
Canonical name	AssociatedPrime
Date of creation	2013-03-22 12:01:37
Last modified on	2013-03-22 12:01:37
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	10
Author	rspuzio (6075)
Entry type	Definition
Classification	msc 16D25
Synonym	annihilator prime