Primitive Rings, An Introduction

Introduction

A ring R is left semiprimitive if there exists a left semisimple faithful module M.

Recall that the jacobson radical is the intersection of the left maximal ideals. These ideals are, in turn, the annihilators of the left simple modules. And M is the direct sum of simple modules.

Assume M is faithful. In the noncommutative world, H×M is the tensor product between the left module M and a right or two sided ideal H, giving an abelian group. Since M is faithful, the group is nonzero.

Consider J×M, where J is the jacobson radical. Since J kills every simple submodule of M, J kills M. This is a contradiction, hence J = 0 and R is jacobson semisimple.

Conversely, assume J = 0. Let M be the direct sum of all left simple modules. Apply any x, and x will lie outside of J, and outside of some maximal left ideal. Thus x doesnot kill one of the components of M, and the result is nonzero. M is faithful. R is left semiprimitive iff it is jacobson semisimple.

By symmetry, R is right semiprimitive iff it is jacobson semisimple. Left semiprimitive = right semiprimitive, and we may as well call it a semiprimitive ring. That has become the accepted terminology. Of course, we don't spend much time talking about semiprimitive rings, since they are synonymous with jacobson semisimple.

Primitive Rings

R is left primitive if there exists a left simple faithful module M. Since primitive implies semiprimitive, R is jacobson semisimple.

Left and right primitive are different; we'll prove that later.

Ideals

An ideal is semiprimitive or left primitive if the quotient ring is same. Note that we are talking about a two sided ideal, so that the quotient makes sense.

A semiprimitive ideal has to contain jac(R).

The ideal H is semiprimitive or left primitive iff H is precisely the annihilator of a semisimple or simple R module. Mod out by H, and M becomes a well defined R/H module, such that nothing kills M.

In general, primitive rings abound, since they can be created from any ring. Mod out by a maximal left ideal, and find a simple left R module M. Let H be the anihilator of M. Now R/H is a left primitive ring.

Commutative Ring

If R is a field, let M = R, a simple faithful R module, and R becomes a primitive ring.

Conversely, let R be a primitive commutative ring. The simple module M is killed by a maximal ideal, and since M is faithful, this ideal has to be 0. That makes R a field.

A commutative ring is primitive iff it is a field. Things just aren't interesting until R is noncommutative.

Endomorphisms Build a Primitive Ring

Let V be a right K vector space, for some division ring K. Let R be the ring of K endomorphisms of V, acting on the left. Note that V is a left R module; we will use V to prove R is primitive.

An element of R is a square matrix, whose rows and columns correspond to the basis of V. The first column is the image of B₁, and so on. Each column has finitely many nonzero entries, thus building an element of V. Endomorphisms are added and multiplied using standard matrix operations. Unless the dimension is 1 and K is a field, R is a noncommutative ring.

Only the 0 endomorphism takes all of V to 0; hence V is faithful.

Start with any vector c in V, and find an endomorphism that maps all the basis elements of c to 0, except one. The result is a single basis element b_i. This can be mapped to any other basis element, and after that, all linear combinations are possible. Thus V is a cyclic R module, where any vector in V acts as generator. This makes V a simple left R module. Since V is simple and faithful, R is left primitive.

If the dimension of V is finite, R is a simple ring. In contrast, an infinite dimension allows for an intermediate ideal H, consisting of matrices with finitely many nonzero rows.