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# associated prime

Let $R$ be a ring, and let $M$ be an $R$-module. A prime ideal $P$ of $R$ is an annihilator prime 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 $(R/P)$-module.

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

Synonym:

annihilator prime

Type of Math Object:

Definition

Major Section:

Reference

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## Mathematics Subject Classification

16D25*no label found*

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