View source for Primitive ring

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''right primitive ring''

An associative ring (cf. [[Associative rings and algebras|Associative rings and algebras]]) with a right faithful [[Irreducible module|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|field]]. Every semi-simple (in the sense of the [[Jacobson radical|Jacobson radical]]) ring is a subdirect product of primitive rings. A [[Simple ring|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|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.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Jacobson,   "Structure of rings" , Amer. Math. Soc.  (1956)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.N. Herstein,   "Noncommutative rings" , Math. Assoc. Amer.  (1968)</TD></TR></table>



====Comments====
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|socle]] which can be described completely [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.H. Rowen,   "Ring theory" , '''I, II''' , Acad. Press  (1988)</TD></TR></table>