Difference between revisions of "Primitive ring"
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''right primitive ring'' | ''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. | 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 | + | 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==== | ====References==== | ||
Latest revision as of 18:10, 11 April 2014
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.
References
| [1] | N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956) |
| [2] | I.N. Herstein, "Noncommutative rings" , Math. Assoc. Amer. (1968) |
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 which can be described completely [a1].
References
| [a1] | L.H. Rowen, "Ring theory" , I, II , Acad. Press (1988) |
Primitive ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Primitive_ring&oldid=12714