Louis Rowen

Some results on the center of a ring with polynomial identity

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 Wedderburns theorem (any semiprimitive ring with polynomial identity, whose center is a field, is simple) and a strengthening of Posners 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 Formaneks 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.

Group graded rings

Let R be a group graded ring . The map ( , ): R × R →1 defined by: (x,y) = (xy)1 , is an inner product on R. In this paper we investigate aspects of nondegeneracy of the product, which is a generalization of the notion of strongly G —graded rings,introduced by Dade. We show that various chain conditions are satisfied by R if and only if they are satisfied by R1 , and that when R1 is simple artinian, then R is a crossed product R1 * G. We give conditions for simple R-modules to be completely reducible R1 -modules . Finally, we prove an incomparability theorem,when G is finite abelian.

Division algebras of degree 4 and 8 with involution

We develop necessary and sufficient conditions for central simple algebras to have involutions of the first kind, and to be tensor products of quaternion subalgebras. The theory is then applied to give an example of a division algebra of degree 8 with involution (of the first kind), without quaternion subalgebras, answering an old question of Albert; another example is a division algebra of degree 4 with involution (*) has no (*)-invariant quaternion subalgebras.

The Tropical Rank of a Tropical Matrix

In this article, we develop further the theory of matrices over the extended tropical semiring. We introduce the notion of tropical linear dependence, enabling us to define matrix rank in a sense that coincides with the notions of tropical nonsingularity and invertibility.

Central simple algebras

Wedderburn’s factorization of polynomials over division rings is refined and used to prove that every central division algebra of degree 8, with involution, has a maximal subfield which is a Galois extension of the center (with Galois group Z2⊕Z2⊕Z2). The same proof, for an arbitrary central division algebra of degree 4, gives an explicit construction of a maximal subfield which is a Galois extension of the center, with Galois group Z2⊕Z2. Use is made of the generic division algebras, with and without involution.

General polynomial identities. II

Abstract A generalized polynomial is a formal polynomial with coefficients in a noncommutative ring (so that the coefficients are interspersed with the indeterminates); a generalized monomial of a generalized polynomial is a sum of all monomials in which the indeterminates occur in the same order. A generalized polynomial identity (GI) of R is R-proper if one of its generalized monomials is not a GI of R . The two main results of this paper are: (i) Any GI of a primitive ring P is P soc P -improper, where soc P is the socle of P ; (ii) if R satisfies a GI that is proper for every nonzero homomorphic image of R , then R is a PI -algebra (in the usual sense). These theorems have several immediate applications: 1. (i) If some element r in R is a root of an algebraic equation f ( λ ) = 0 with f ′( r ) invertible, f ′ the formal derivative of f , and if the centralizer of r is a PI -algebra, then R is a PI -algebra (most of this was already known by M. Smith); 2. (ii) If R has two left ideals A and B that are PI -algebras, then A + B is a PI -algebra. The main results of the paper are also given in the context of rings with involution.

Dihedral algebras are cyclic

In his book [1], Albert has a proof that every division algebra of degree 3 is cyclic. In this paper we will generalize this result, and derive the theorem below. Our argument is very close to that of Albert, and arose as part of a close examination of his proof. Fix n to be an odd positive integer, and F a field of characteristic prime to n. Denote by Dn the dihedral group of order 2n. We assume the reader is familiar with the basics of the theory of finite dimensional simple algebras as presented, for example, in Alberts book.

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

Let

COMPLETIONS, REVERSALS, AND DUALITY FOR TROPICAL VARIETIES

p

Fields of Definition for Division Algebras

be a multilinear polynomial in several non-commuting variables with coefficients in a quadratically closed field