Primitive ring


Primitive ring

In mathematics, especially in the area of abstract algebra known as ring theory, the concept of left primitive ring generalizes that of matrix algebra. Every matrix ring is the endomorphism ring of a finite dimensional vector space, but a primitive ring is a special sort of subring of the endomorphism ring of a possibly infinite dimensional vector space. Many important rings are primitive, include non-commutative polynomial rings and simple artinian rings.

Definition

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.

By the Jacobson density theorem, a ring is left primitive if and only if it is isomorphic to a dense ring of endomorphisms of a right vector space over a division ring.

A commutative ring is left primitive if and only if it is a field.

A left artinian ring is left primitive if and only if it is simple if and only if it is prime.

A ring is left primitive if and only if it is prime and has a faithful left module of finite length.

Examples

Many important rings are primitive, including non-commutative polynomial rings, and characteristic zero Weyl algebras.

Properties

Every simple ring "R" is both left and right primitive. However, a simple non-unital ring, may not be primitive. To construct a faithful simple left "R"-module for a given simple ring "R", one first finds (using Zorn's lemma and the fact that "R" has a multiplicative identity) a maximal left ideal "M" in "R". The quotient module "R"/"M" is a simple left "R"-module; its annihilator is a two-sided ideal in "R", and "R" being a simple ring implies that this annihilator is {0} and therefore "R"/"M" is a faithful left "R"-module.

As a consequence of the Jacobson Density Theorem, every primitive ring is a dense subring of the ring of linear transformations of a vector space over a division ring. For left primitive rings, these linear transformations act on the left, and for right primitive rings, they act on the right.

Conversely, it is easy to see that every dense subring of the ring of linear transformations of a vector space over a division ring is primitive. Thus, the theorem completely characterizes primitive rings.

A ring theoretic characterization of left primitive rings is as follows: a ring is left primitive if and only if there is a maximal left ideal whose right core is zero. The dual definition is valid for right primitive rings.

There are primitive rings which are not simple. In particular, the ring of all linear transformations of an infinite dimensional vector space over a division ring is primitive, but is not simple as the set of finite rank linear transformations is a two sided ideal.

There are rings which are primitive on one side but not on the other. The first example was constructed by George M. Bergman in 1964.

References

* Bergman, George M. "A ring primitive on the right but not on the left." Proc. Amer. Math. Soc. 15 (1964) pp. 473-475. Errata in pp. 1000.


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