# 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 $P={\rm ann}(X)$, the annihilator of some nonzero submodule $X$ of $M$.

Note that if this is the case, then the module ${\rm ann}_{M}(P)$ 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 AssociatedPrime 2013-03-22 12:01:37 2013-03-22 12:01:37 rspuzio (6075) rspuzio (6075) 10 rspuzio (6075) Definition msc 16D25 annihilator prime