Primitive ring

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.

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Semi-simple rings in the sense of the Jacobson radical are now called semi-primitive rings. Primitive rings with polynomial identities are central simple finite-dimensional algebras. Primitive rings with minimal one-sided ideals have a socle which can be described completely [a1].

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