Primitive ring

In the branch of abstract algebra known as ring theory, a left primitive ring is a ring which has a faithful simple left module. Well known examples include endomorphism rings of vector spaces and Weyl algebras over fields of characteristic zero. In the branch of abstract algebra known as ring theory, a left primitive ring is a ring which has a faithful simple left module. Well known examples include endomorphism rings of vector spaces and Weyl algebras over fields of characteristic zero. A ring R is said to be a left primitive ring if and only if it has a faithful simple left R-module. A right primitive ring is defined similarly with right R-modules. There are rings which are primitive on one side but not on the other. The first example was constructed by George M. Bergman in (Bergman 1964). Another example found by Jategaonkar showing the distinction can be found in (Rowen & 1988, p.159) An internal characterization of left primitive rings is as follows: a ring is left primitive if and only if there is a maximal left ideal containing no nonzero two-sided ideals. The analogous definition for right primitive rings is also valid. The structure of left primitive rings is completely determined by the Jacobson density theorem: A ring is left primitive if and only if it is isomorphic to a dense subring of the ring of endomorphisms of a left vector space over a division ring. Another equivalent definition states that a ring is left primitive if and only if it is a prime ring with a faithful left module of finite length (Lam 2001, Ex. 11.19, p. 191). One sided primitive rings are both semiprimitive rings and prime rings. Since the ring product of two or more nonzero rings is not prime, it is clear that the product of primitive rings is never primitive. For a left Artinian ring, it is known that the conditions 'left primitive', 'right primitive', 'prime', and 'simple' are all equivalent, and in this case it is a semisimple ring isomorphic to a square matrix ring over a division ring. More generally, in any ring with a minimal one sided ideal, 'left primitive'='right primitive'='prime'. A commutative ring is left primitive if and only if it is a field. Being left primitive is a Morita invariant property.