Introduction. The purpose of this paper is to provide a fresh outlook to various questions on rings with polynomial identity by examining the centers of such rings. This approach yields the interesting result that any nonzero ideal of a semiprime ring with polynomial identity intersects the center nontrivially (Theorem 2). There are at least two interesting consequences to Theorem 2: a generalization of Wedderburn's theorem (any semiprimitive ring with polynomial identity, whose center is a field, is simple) and a strengthening of Posner's theorem [1] (any prime ring with a polynomial identity has a simple ring of quotients whose center is the quotient field of the center of the prime ring). The proofs are elementary modulo Jacobson [3]. Of course rings are not necessarily commutative and for the sake of simplicity we assume a unit 1. The key argument in this paper is an application of Formanek's central polynomials for matrix algebras over a field, whose important properties are [2] : Let Mn be an n x n matrix algebra over an arbitrary field. Then there exists a polynomial gn(Xl9.. .,Xm) which has coefficients in Z; is homogeneous (degree > 0) in every variable and linear in all but the first variable; takes values in the center for every specialization in Mn; and is nonvanishing for some specialization.

background or methods

/pdf/some-results-on-the-center-of-a-ring-with-polynomial-22giuh2vsz.pdf

Central polynomials for matrix rings

Some results on the center of a ring with polynomial identity

/pdf/nil-and-power-central-polynomials-in-rings-26l2i0n3ug.pdf

Nil and power-central polynomials in rings

/pdf/on-the-von-neumann-regularity-of-rings-with-regular-prime-57d9tnc6kj.pdf

On the von Neumann regularity of rings with regular prime factor rings

Let R be a ring, K a field, and n a natural number. We will be concerned with the following type of questions: (a) Classify the representations of R in (hO, (or n dimensional representations). (b) Classify n dimensional representations up to the natural equivalence. If ~bt, ~b2 : R ~ (K), we say ~bx is equivalent to ~b 2 if there is a K automorphism 7 of (K), such that 7q~1 = 4)2. (c) Classify equivalence classes of irreducible and semisimple representations. For the sake of simplicity we will restrict ourselves mainly to the case that K is algebraically closed and R is a finitely generated K algebra. In fact, for most of this paper, we will assume that R = K{xa, ..., x,) is a free algebra; at the end we will deduce from the results in this ease the more general theorems for not necessarily free algebras. Before describing the theorems that we will obtain, let us digress in order to motivate the use that we will make of the theory of rings with polynomial identities as a tool for attacking the problems stated before. Let us recall, therefore, some of the basic structural results of this theory and interpret them in terms of representation theory.

background

Finite dimensional representations of algebras

/pdf/the-images-of-non-commutative-polynomials-evaluated-on-2-x-2-29ctkv1i1a.pdf

Central polynomials for matrix rings

Citations

Some results on the center of a ring with polynomial identity

Nil and power-central polynomials in rings

On the von Neumann regularity of rings with regular prime factor rings

Finite dimensional representations of algebras

The images of non-commutative polynomials evaluated on 2 x 2 matrices

References

“Problems in the Theory of Rings” Revisited

On a problem of kaplansky

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