Associative rings and algebras Definition

Primitive ideal

Builds on Ring Two-sided ideal Module over a ring Simple module Primitive ring

Definition

Let $R$ be a ring with identity. A two-sided ideal $P$ of $R$ is a left primitive ideal if there is a simple left $R$ -module $M$ with $P = \operatorname{ann}_R(M) = \set{r \in R : rm = 0 \text{ for all } m \in M}.$

Remarks

Equivalently, $P$ is left primitive if and only if the quotient ring $R/P$ is a left primitive ring. Example: in $\Z$ , for a prime $p$ the ideal $p\Z$ is the annihilator of the simple module $\Z/p\Z$ , so it is primitive. Non-example: the zero ideal of $\Z$ is prime but not primitive, since $\Z/0 \cong \Z$ is not a field. In a commutative ring the primitive ideals are exactly the maximal ideals. The Jacobson radical $J(R)$ is the intersection of all left primitive ideals of $R$ . Right primitive ideals are defined with right modules; the two notions can differ.

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Group concept
Module over a ring definition
Primitive ring concept
Simple module concept
Ring definition
Ideal definition
Two-sided ideal definition
Primitive ideal definition

Tag	`Z50EV9SYNJ`
URI	`m2://Z50EV9SYNJ`
Kind	definition
Writing	W0 stub
MSC	16D60, 16N20
Defines	Primitive ideal
Uses	Ring, Two-sided ideal, Module over a ring, Simple module, Primitive ring

Primitive ideal

Definition

Remarks

Reading path

Further reading

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