Namespaces
Variants
Actions

Primitive ring

From Encyclopedia of Mathematics
Revision as of 18:10, 11 April 2014 by Ivan (talk | contribs) (TeX)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

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)
How to Cite This Entry:
Primitive ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Primitive_ring&oldid=12714
This article was adapted from an original article by K.A. Zhevlakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article