primitive ideal
PrimitiveIdeal
2001-11-24 02:22:45
2002-10-25 18:11:15
Definition
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Let $R$ be a ring, and let $I$ be an ideal of $R$.
We say that $I$ is a {\it left (right) primitive ideal}
if there exists a simple left (right) $R$-module $X$
such that $I$ is the annihilator of $X$ in $R$.
We say that $R$ is a {\it left (right) primitive ring}
if the zero ideal is a left (right) primitive ideal of $R$.
Note that $I$ is a left (right) primitive ideal
if and only if $R/I$ is a left (right) primitive ring.