# Primitive ring

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right primitive ring

An associative ring (cf. Associative rings and algebras) with a right faithful irreducible module. Analogously (using a left irreducible module) one defines a left primitive ring. The classes of right and left primitive rings do not coincide. Every commutative primitive ring is a field. Every semi-simple (in the sense of the Jacobson radical) ring is a subdirect product of primitive rings. A simple ring is either primitive or radical. The primitive rings with non-zero minimal right ideals can be described by a density theorem. The primitive rings with minimum condition for right ideals (i.e. the Artinian primitive rings) are simple.

A ring \$R\$ is primitive if and only if it has a maximal modular right ideal \$I\$ (cf. Modular ideal) that does not contain any two-sided ideal of \$R\$ distinct from the zero ideal. This property can be taken as the definition of a primitive ring in the class of non-associative rings.

How to Cite This Entry:
Primitive ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Primitive_ring&oldid=31545
This article was adapted from an original article by K.A. Zhevlakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article