Amiram Braun

The nilpotency of the radical in a finitely generated P.I. ring

Abstract Let R = Λ { x 1 ,…, x k } be a p.i. ring, satisfying a monic polynomial identity (one of its coefficients is ± 1), where Λ is a central noetherian subring. It is proven that N ( R ), the nil radical of R , is nilpotent. As a corollary, by taking Λ = F , a field, we settle affirmatively the open problem posed in (C. Procesi, “Rings with Polynomial Identities,” p. 186, Marcel Dekker, New York, 1973). We prove: “The Jacobson radical of a finitely generated p.i. algebra is nilpotent.”

Integrality for PI-rings

Abstract In this paper, we study (Schelter) integral extensions of PI-rings. We prove in particular lying over, going up, and incomparability for prime ideals. A major result is transitivity of integrality: If R⊆S⊆B are PI-rings such that B is integral over S and S is integral over R, then B is integral over R. Next, we obtain a powerful criterion for integrality: If S is a prime PI-ring such that its center is integral over a Noetherian subring R of S, then S is integral over R. This allows interesting applications to the theory of finite group actions. Further topics concern Eakin-Nagata type results and embeddings of quotient rings for integral extensions. Finally, we analyze the relationship between module-finite extensions and finitely generated integral extensions, obtaining positive results for affine Noetherian PI-algebras and algebras satisfying certain restrictions on PI-degrees (e.g., algebras of low PI-degree).

On Artin's theorem and Azumaya algebras

A short proof of the existence of central polynomial of matrix rings is given and some of its applications. The following characterization of Azumaya algebras is proved: R is Azumaya iff there exist ai, bi ϵ R, i = 1,…, k with ∑ik = 1 airbi ϵ Z(R), ∀r ϵ R and ∑ aibi = 1. This is proved as a consequence of the following generalization of a theorem due to M. Artin (and generalized by C. Procesi): Let R = Λ{x1,…, xk} be a p.i. ring, Λ a central noetherian subring. Then R is Azumaya iff for every two sided ideal I in R, Z(RI) = Z(R)I ∩ Z(R), where Z(R) denotes the center of R.

Localization, completion and the AR property in Noetherian P.I. rings

Given a prime idealP in a noetherian ringR we examine the following two properties: (1)P is Ore localizable. (2) The completion ofR atP is Noetherian. For rings satisfying the 2nd layer condition a strong connection is discovered between (1) and (2) and consequently questions by Goldie and McConnell are answered. As a corollary we also obtain a new characterization for non-maximal primitive idealP inR to satisfy (1), whereR is the enveloping algebra of complex solvable finite dimensional Lie algebra

P.I. rings and the localization at height 1 prime ideals

LetR be a prime P.I. ring, finitely generated over a central noetherian subring. LetP be a height one prime ideal inR. We establish a finite criteria for the left (right) Ore localizability ofP, providedP/P2 is left (right) finitely generated. This replaces the noetherian assumption onR appearing in [BW], using an entirely different technique.

ON P.I. RING AND STANDARD IDENTITIES

An example is given of a ringR (with 1) satisfying the standard identityS6[x1, ...,x6] butM2(R), the 2 × 2 matrix ring overR, does not satisfyS12[x1, ...,x12]. This is in contrast to the caseR=Mn (F),F a field, where by the Amitsur-Levitzki theoremR satisfiesS2n [x1, ...,x2n] andM2(R) satisfiesS4n [x1, ...,xn].

On bimodules over Noetherian PI rings

Let R be a prime Noetherian PI ring, and let I be an ideal in R satisfying xI ⊆ Ix for some x in R. We prove that xI = Ix. This is obtained as a corollary of a similar more general result, where I can be taken as any finitely generated torsion-free central R-bimodule.

Hereditary and maximal crossed product orders

We first deal with classical crossed products S f * G, where G is a finite group acting on a Dedekind domain S and S G (the G-invariant elements in S) a DVR, admitting a separable residue fields extension. Here f : G x G → S* is a 2-cocycle. We prove that S f * G is hereditary if and only if S/Jac(S) f * G is semi-simple. In particular, the heredity property may hold even when S/S G is not tamely ramified (contradicting standard textbook references). For an arbitrary Krull domain S, we use the above to prove that under the same separability assumption, S f * G is a maximal order if and only if its height one prime ideals are extended from S. We generalize these results by dropping the residual separability assumptions. An application to non-commutative unique factorization rings is also presented.

Smooth PI algebras with finite divisor class group

We have shown in an earlier paper that the divisor class group of the centre of a smooth PI algebra with trivial K(0) is a torsion group of finite exponent. We show here that this group need not be finite even in the affine case. Our example is an Azumaya algebra of global dimension 2. We also provide a positive result in a special case.

Localization in noetherian p.i. rings as a finite criterion

Several equivalent (finite) conditions for the localizability of a prime ideal in a noetherian prime p.i. ring are given. It is used to connect localizability to the curve criterion. Another application is to some integrality questions.