Goldie Ring Research Articles

Classical quotient rings

Throughout R is a ring with right singular ideal Z ( R ) Z(R) . A right ideal K of R is rationally closed if x − 1 K = { y ∈ R : x y ∈ K } {x^{ - 1}}K = \{ y \in R:xy \in K\} is not a dense right ideal for all x ∈ R − K x \in R - K . A ring R is a Cl-ring if R is (Goldie) right finite dimensional, R / Z ( R ) R/Z(R) is semiprime, Z ( R ) Z(R) is rationally closed, and Z ( R ) Z(R) contains no closed uniform right ideals. These rings include the quasi-Frobenius rings as well as the semiprime Goldie rings. The commutative Cl-rings have Cl-classical quotient rings. The injective ones are congenerator rings. In what follows, R is a Cl-ring. A dense right ideal of R contains a right nonzero divisor. If R satisfies the minimum condition on rationally closed right ideals then R has a classical Artinian quotient ring. The complete right quotient ring Q (also called the Johnson-Utumi maximal quotient ring) of R is a Cl-ring. If R has the additional property that bR is dense whenever b is a right nonzero divisor, then Q is classical. If Q is injective, then Q is classical.

Goldie-like conditions on Jordan matrix rings

In this paper Goldie-like conditions are put on a Jordan matrix ring J = H ( R n , γ a ) 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.

A theorem on Noetherian hereditary rings

It is shown (Theorem 2) that a semi-prime, left noetherian, left hereditary, two-sided Goldie ring is right noetherian if and only if the right module (Q/R) φ R contains a copy of every simple right iέ-module, where Q is the classical quotient ring of R. Theorem 5 gives several necessary and sufficient conditions for a semi-prime principal left ideal ring which is right Goldie to be a principal right ideal ring. Among these is that R/A must be artinian for every essential left ideal A. It is known that a two-sided noetherian semi-prime ring is principal on the left if and only if it is principal on the right. On the other hand, if one drops the ascending chain condition on the right side of R, examples are known of principal left ideal domains (p.l.i. domains) which are not right principal. But, if we require that they be right Ore as well, things may be better.

On classical quotients of polynomial identity rings with involution

Let ( R , ∗ ) (R, \ast ) denote a ring R R with involution ( ∗ ) ( \ast ) , where “involution” means “ anti - automorphism of order ≦ two {\text {anti - automorphism of order}} \leqq {\text { two}} ". We can specialize many ring-theoretical concepts to rings with involution; in particular an ideal of ( R , ∗ ) (R, \ast ) is an ideal of R R stable under ( ∗ ) ( \ast ) , and the center of ( R , ∗ ) (R, \ast ) is the set of central elements of R R which are fixed under ( ∗ ) ( \ast ) . Then we say ( R , ∗ ) (R, \ast ) is prime when the product of any two nonzero ideals of ( R , ∗ ) (R, \ast ) is nonzero; similarly ( R , ∗ ) (R, \ast ) is semiprime when any power of a nonzero ideal of ( R , ∗ ) (R, \ast ) is nonzero. The main result of this paper is a strong analogue to Posner’s theorem [5], namely that any prime ( R , ∗ ) (R, \ast ) with polynomial identity has a ring of quotients R T {R_T} , formed merely by adjoining inverses of nonzero elements of the center of ( R , ∗ ) (R, \ast ) . This quotient ring ( R T , ∗ ) ({R_T}, \ast ) is simple and finite dimensional over its center. An extension of these results to semiprime Goldie rings with polynomial identity is given.

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

Classical quotient rings of PWD’s

Piecewise domains which are right orders in semiprimary rings are characterized. An example is given showing the result obtained is best possible. A further example is obtained of a prime right Goldie ring possessing a regular element which becomes a left zero divisor in some prime overring. This example leads to the construction of a PWD R not satisfying the regularity condition, but for which R/N(R) is right Goldie. Introduction. The principle result of this note is a characterization of PWD's (piecewise domains) which are right orders in semiprimary rings. The main device employed here is L. W. Small's concept of exhaustive set. Probably the most important consequence of the characterization is that a right noetherian PWD is a right order in a right artinian PWD. In the more general setting our result is less satisfactory because an assumption is made about each of the factor rings R/Ti(R) (see the theorem in ?1). However, an example given at the end of ?1 shows that this is unavoidable: There we exhibit a right Goldie PWD R having R/N(R) right Goldie which is not a right order in a semiprimary ring. In ?2, as a problem closely related to the existence of classical quotient rings, we study the regularity condition in PWD's. Necessary and sufficient conditions are obtained for a PWD R to satisfy the regularity condition in terms of the torsion-freeness of R as an R/N(R)-module. It is shown that any PWD R with R/N(R) right Goldie satisfies a certain one-sided regularity condition. That the regularity condition then need not be implied is demonstrated by an example. Since this example is surely of more general interest than its application, we single it out here: A right Ore domain D is constructed so that for some elements a, b, c, d E D, the equations ax+by=O, cx+dy=O have a nontrivial solution in a domain E extending D, but no nontrivial solution in D. (This means that the prime right Goldie ring D2 contains a regular element which becomes a left zero divisor in the prime overring E2.) It is amusing to note that there Received by the editors September 7, 1971. AMS 1970 subject classifications. Primary 16A08; Secondary 16A02, 15A06. Kev words and phrases. Piecewise domain, essentiality condition, exhaustive set, order, classical quotient ring, Goldie ring, principal right ideals projective, torsionfree, left regularity condition, Ore domain, linear equations. ( American Mathematical Society 1972

Semi-Prime Right Goldie Rings which are Direct Sums of Uniform Right Ideals

Semi-Prime Right Goldie Rings which are Direct Sums of Uniform Right Ideals

On the Nilpotency of Nil Subrings

A famous theorem of Levitzki states that in a left Noetherian ring each nil left ideal is nilpotent. Lanski [5] has extended Levitzki's theorem by proving that in a left Goldie ring each nil subring is nilpotent. Another important theorem in this area which is due to Herstein and Small [3] states that if a ring satisfies the ascending chain condition on both left and right annihilators, then each nil subring is nilpotent. We give a short proof of a theorem (Theorem 1.6) which yields both Lanski's theorem and Herstein- Small's theorem. We make use of the ascending chain condition on principal left annihilators in order to obtain, at an intermediate step, a theorem (Theorem 1.1) which produces sufficient conditions for a nil subring to be left T-nilpotent.

On tensor products of semiprime algebras which are goldie rings

On tensor products of semiprime algebras which are goldie rings

Nil Subrings of Goldie Rings are Nilpotent

Herstein and Small have shown (1) that nil rings which satisfy certain chain conditions are nilpotent. In particular, this is true for nil (left) Goldie rings. The result obtained here is a generalization of their result to the case of any nil subring of a Goldie ring.Definition. Lis a left annihilator in the ring R if there exists a subset S ⊂ R with L = {x∈ R|xS= 0}. In this case we write L= l(S). A right annihilator K = r(S) is defined similarly.Definition. A ring R satisfies the ascending chain condition on left annihilators if any ascending chain of left annihilators terminates at some point. We recall the well-known fact that this condition is inherited by subrings.Definition. R is a Goldie ring if R has no infinite direct sum of left ideals and has the ascending chain condition on left annihilators.

On a theorem of Goldie

Abstract : The theorems proven are: (1) Let R be a semiprime (left) Goldie ring; then R has a left quotient ring Q; and (2) Q is a semisimple ring satisfying the descending chain condition on left ideals.