Ordered ∗ -Rings

has characterized those rings with involution J, with 1, and such that every symmetric element has an inverse under the additional hypothesis that S~~, the subring generated by ,S (the set of symmetric elements), is A, the ring itself.In this paper we raise the question as to when S~ = A under a weaker hypothesis and prove the following theorem.(A henceforth is such that 2A =A and 2x=0 implies x=0 for any xEA.)Theorem 1.Let A be a ring with 1 and suppose that S is a simple Jordan ring under the Jordan multiplication, s o t = st+ts for all s and t in S. Then either S~ = A or SEZ, the center of A, or K, the set o/ skew elements, is an ideal o/ A with K2EZ, K3 = (8), and S is an associative ring under Jordan multiplication.We note that Osborn proves that under his hypothesis, either A is a division ring; the direct sum of two division rings which are antiisomorphic, and / interchanges the summands; or 51 is a field under Jordan multiplication, K is an ideal of A where K2 = (d).A corollary to our result is Corollary 1.Let A be as in Theorem 1. Suppose /urther that A contains no nilpotent ideals.Then either A is simple or A is a direct sum o/ two simple rings which are anti-isomorphic and such that J interchanges the direct summands.

Let R be a 2-torsion free commutative ring with involution, and δ a nonzero derivation of R. Let S be the set of symmetric elements in R, and let K be the set of anti-symmetric elements in R. In this article, we investigate the semiprimeness of the Lie rings Sδ when δ is symmetric and Kδ when δ is anti-symmetric.

Let RG denote the group ring of a group G over a commutative ring with unity R. Given a homomorphism σ:G → {± 1} and an involution ϕ of the group G, an oriented invoultion σϕ of RG is defined in a natural way. We characterize when the set of symmetric elements under this involution is a subring. This gives a unified setting for earlier work of several authors.

Herstein's Lie theory revisited

Let R be a primitive ring with involution *. Thus R may be considered as an irreducible ring of endomorphisms of an additive abelian group V, so that D=HomR(V, V) is a division ring. Let C be the center of D. We shall furthermore assume that CRCR. It can be shown that an involution -y--+is induced in C which has the property that 1yx = (tyx*)* for all xER. The involution * is of the first kind if -y-+-7 is the identity mapping and is of the second kind if there is a -y Oe C such that fy = -y. The set of symmetric elements of R will be denoted by S. We now assume that S satisfies a nontrivial generalized polynomial identity over C (in the sense of Amitsur). This means that there exists a nonzero element f(xl, x2, * * *, x.) in the so-called C-universal product R(x) of the C-algebra R and the free C-algebra C[X,, x2,* * *, x,, ... ] in noncommuting indeterminants xi, X2, *. * Xn, ...* such thatf(s1, S2, * * *, sn) = 0 for all SI S2, *.*.*, SnCS. For more precise details concerning the above notions we refer the reader to [1, ?4], and [2, ?3]. The usual linearization process may be used so that we may assume without loss of generality that S satisfies a nontrivial generalized (homogeneous) multilinear identity of degree n in xi, x2, ,n:

In the first part of this work, we study the Gelfand-Kirillov dimension of symmetric algebras H(A, * ) and associative algebras with involution (A, * ). In addition, we make a contribution by relating the Gelfand-Kirillov dimensions of H(A, * ) and (A, * ). In the second part, we prove that the set of symmetric elements of an alternative algebra with involution is finitely generated, which is analogous to Osborn's theorem for associative algebras. In addition, we consider two Jordan algebras related with the first Weyl algebra A 1 and find their representations in terms of generators and relations. Finally, we study locally nilpotent derivations of free associative algebra K x, y (K field of characteristic zero) and show that they are triangularizable. As an application, we give new proof that the automorphisms of K x, y are tame.