Prime Goldie Ring Research Articles

Monomial conditions on rings

Drazin introduced the notion ofpivotal monomial, a condition on the evaluations of monomials in a ring, and characterized simple artinian rings as those primitive rings which have pivotal monomials. In this paper we consider monomial conditions related to pivotal monomials. The two major results are a characterization of prime Goldie rings in terms of pivotal monomials, and a characterization of the socle of a primitive ring in terms of generalized pivotal monomials.

Strongly prime rings

A ring R is (right) strongly prime (SP) if every nonzero twosided ideal contains a finite set whose right annihilator is zero. Examples are domains, prime Goldie rings and simple rings; however, this notion is asymmetric and a right but not left SP ring is exhibited. All SP rings are prime, and every prime ring may be embedded in an SP ring. SP rings are nonsingular, and a regular SP ring is simple; since faithful rings of quotients of SP rings are SP, the complete ring of quotients of an SP ring is simple. All SP rings are coefficient rings for some primitive group ring (a generalization of a result proved for domains by Formanek), and this was the initial motivation for their study. If the group ring RG is SP, then R is SP and G contains no nontrivial locally finite normal subgroups. Coincidentally, SP rings coincide with the ATF rings of Rubin, and so every SP ring has a unique maximal proper torsion theory, and (0) and R are the only torsion ideals.( 1 ^{1} ) A list of questions is appended.

Goldie-Like Conditions on Jordan Matrix Rings

In this paper Goldie-like conditions are put on a Jordan matrix ring $J = H({R_n},{\gamma _a})$ which are necessary and sufficient for R to be a $\ast$-prime Goldie ring or a Cayley-Dickson ring. Existing theory is then used to obtain a Jordan ring of quotients for J.

Prime rings with involution and generalized polynomial identities

Our main purpose in writing this paper is to prove that if R is a prime ring with involution whose symmetric elements satisfy a generalized polynomial identity over the extended centroid C of R, then the central closure A = RC + C must in fact be a primitive ring with a minimal right ideal eA such that eAe is a finite dimensional division algebra over C. This generalizes a previous theorem of ours [8], in which we obtained the above result for the case where R was assumed to be primitive. It also generalizes a recent result of Skinner [lo, Theorem 5.11, where the above result was obtained for the case where R was a prime Goldie ring. In the course of proving this theorem, and without digressing too much from our main goal, we rework and combine the techniques of Amitsur in [l] and [2] with those of ours in [7], so as to obtain the various structure theorems (due to Amitsur, Posner, Herstein, Kaplansky, and the author) on simple, primitive, and prime rings (with involution) whose (symmetric) elements satisfy a (generalized) polynomial identity. We hope this attempt to give a more or less unified approach to a group of theorems, the existing proofs of which are not for the most part too closely related, will be of general interest. In Section 2, we recall the notion of extended centroid of a prime ring and discuss some of its key properties (Theorems 2.1-2.4). Next, putting together some ideas of Amitsur, we give a particular way of embedding prime rings in primitive rings (Theorems 2.5-2.8). Finally, we show how information about primitive rings can be pulled back to prime rings (Theorems 2.9-2.10). In Section 3, we recall the notion of generalized multilinear identity (GMI) and show (Theorems 3.1-3.2) that GMI’s are carried over by the aforementioned embedding of prime rings in primitive rings. A fundamental result of Amitsur (Theorem 3.3) together with its corollary, Theorem 3.5, are applicable to primitive rings and say in effect that some nonzero linear