Polynomial Identity Rings Research Articles

Additive decomposition of ideals

In this paper, we investigate decomposition of (one-sided) ideals of a unital ring [Formula: see text] as a sum of two (one-sided) ideals, each being idempotent, nil, nilpotent, T-nilpotent, or a direct summand of [Formula: see text]. Among other characterizations, we prove that in a polynomial identity ring every (one-sided) ideal is a sum of a nil (one-sided) ideal and an idempotent (one-sided) ideal if and only if the Jacobson radical [Formula: see text] of [Formula: see text] is nil and [Formula: see text] is von Neumann regular. As a special case, these conditions for a commutative ring [Formula: see text] are equivalent to [Formula: see text] having zero Krull dimension. While assuming Köthe’s conjecture in several occasions to be true, we also raise a question, the affirmative answer to which leads to the truth of the conjecture.

A Prime Ideal Principle for Two-Sided Ideals

Many classical ring-theoretic results state that an ideal that is maximal with respect to satisfying a special property must be prime. We present a “Prime Ideal Principle” that gives a uniform method of proving such facts, generalizing the Prime Ideal Principle for commutative rings due to T. Y. Lam and the author. Old and new “maximal implies prime” results are presented, with results touching on annihilator ideals, polynomial identity rings, the Artin–Rees property, Dedekind-finite rings, principal ideals generated by normal elements, strongly noetherian algebras, and just infinite algebras.

Differential smoothness of affine Hopf algebras of Gelfand–Kirillov dimension two

Two-dimensional integrable differential calculi for classes of Ore exten- sions of the polynomial ring and the Laurent polynomial ring in one variable are con- structed. Thus it is concluded that all affine pointed Hopf domains ofGelfand-Kirillov dimension two which are not polynomial identity rings are differentially smooth.

Polynomial identity rings as rings of functions, II

In characteristic zero, Zinovy Reichstein and the author generalized the usual relationship between irreducible Zariski closed subsets of the affine space, their defining ideals, coordinate rings, and function fields, to a non-commutative setting, where “varieties” carry a PGLn-action, regular and rational “functions” on them are matrix-valued, “coordinate rings” are prime polynomial identity algebras, and “function fields” are central simple algebras of degree n. In the present paper, much of this is extended to prime characteristic. In addition, a mistake in the earlier paper is corrected. One of the results is that the finitely generated prime PI-algebras of degree n are precisely the rings that arise as “coordinate rings” of “n-varieties” in this setting. For n=1 the definitions and results reduce to those of classical affine algebraic geometry.

Polynomial identity rings as rings of functions

We generalize the usual relationship between irreducible Zariski closed subsets of the affine space, their defining ideals, coordinate rings, and function fields, to a non-commutative setting, where “varieties” carry a PGL n -action, regular and rational “functions” on them are matrix-valued, “coordinate rings” are prime polynomial identity algebras, and “function fields” are central simple algebras of degree n. In particular, a prime polynomial identity algebra of degree n is finitely generated if and only if it arises as the “coordinate ring” of a “variety” in this setting. For n = 1 our definitions and results reduce to those of classical affine algebraic geometry.

Book Review: Polynomial identity rings

Book Review: Polynomial identity rings

A Structure Theorem for Noetherian P.I. Rings with Global Dimension Two

LetAbe the Artin radical of a Noetherian ringRof global dimension two. We show thatA=ReRwhereeis an idempotent;Acontains a heredity chain of ideals and the global dimensions of the ringsR/AandeRecannot exceed two. Assume further thanRis a polynomial identity ring. LetPbe a minimal prime ideal ofR. ThenP=P2and the global dimension ofR/Pis also bounded by two. In particular, if the Krull dimension ofR/Pequals two for all minimal primesPthenRis a semiprime ring. In general, every clique of prime ideals inRis finite and in the affine caseRis a finite module over a commutative affine subring. Additionally, whenA=0, the ringRhas an Artinian quotient ring and we provide a structure theorem which shows thatRis obtained by a certain subidealizing process carried out on rings involving Dedekind prime rings and other homologically homogeneous rings.

Prime rings with PI rings of constants

It is shown that if the ring of constants of a restricted differential lie algebra with a quasi-Frobenius inner part satisfies a polynomial identity (PI) then the original prime ring has a generalized polynomial identity (GPI). If additionally the ring of constants is semiprime then the original ring is PI. The case of a non-quasi-Frobenius inner part is also considered.

One sided invertibility and localisation II

The aim of this paper is to generalise the results of [7] from the prime to the semiprime case. It was shown, for instance, that if M is the annihilator of a simple right module S of projective dimension 1 over a Noetherian prime polynomial identity (PI) ring R then M is either an invertible ideal or an idempotent ideal [7, Proposition 4.2]. One of the main applications of this result was that a prime Noetherian affine PI ring of global dimension less than or equal to 2 is a finite module over its centre. It turns out that this theorem is valid more generally when the ring is semiprime [1, Theorem A]. Clearly this requires [7, Proposition 4.2] also to be strengthened to the semiprime case. We do this by showing that a right invertible maximal ideal in a semiprime Noetherian PI ring is also left invertible (Theorem 3.5).

A test for finite representation type

Let R be a left pure-semisimple ring. It is proved that if R has self-duality or if R is a polynomial identity ring, then R is of finite representation type. If there exists an example of a left pure-semisimple ring which is not of finite representation type, we show that then there exists an example which is hereditary, but not right artinian. We reduce the question of the existence of such an example to a problem regarding simple bimodules over division rings.

One sided invertibility and localisation

In general, a prime ideal P of a prime Noetherian ring need not be classically localisable. Since such a localisation, when it does exist, is a striking property; sufficiency criteria which guarantee it are worthy of careful study. One such condition which ensures localisation is when P is an invertible ideal [5, Theorem 1.3]. The known proofs of this result utilise both the left as well as the right invertiblity of P. Such a requirement is, in practice, somewhat restrictive. There are many occasions such as when a product of prime ideals is invertible [6] or when a non-idempotent maximal ideal is known to be projective only on one side [2], when the assumptions lead to invertibilty also on just one side. Our main purpose here is to show that in the context of Noetherian prime polynomial identity rings, this one-sided assumption is enough to ensure classical localisation [Theorem 3.5]. Consequently, if a maximal ideal in such a ring is invertible on one side then it is invertible on both sides [Proposition 4.1]. This result plays a crucial role in [2]. As a further application we show that for polynomial identity rings the definition of a unique factorisation ring is left-right symmetric [Theorem 4.4].

Varieties of rings with definable principal congruences

A variety V \mathcal {V} of rings has definable principal congruences (DPC) if there is a first order sentence defining principal two-sided ideals for all rings in V \mathcal {V} . The key result is that for any ring R R , V ( M n ( R ) ) V({M_n}(R)) does not have DPC if n ⩾ 2 n \geqslant 2 . This allows us to show that if V ( R ) V(R) has DPC, then R R is a polynomial identity ring. Results from the theory of PI rings are used to prove that for a semiprime ring R R , V ( R ) V(R) has DPC if and only if R R is commutative. An example of a finite, local, noncommutative ring R R with V ( R ) V(R) having DPC is given.

Retracts and Injectives

AbstractEmbedding theorems are employed to show that many important categories do not possess non-trivial retracts or injectives. E.g., the categories of monoids, groups, rings, rings with unity, polynomial identity rings, nilpotent groups, solvable groups, and several varieties of groups.

Affine polynomial identity rings and their generalizations

Affine polynomial identity rings and their generalizations

A Version of Zariski's Main Theorem for Polynomial Identity Rings

A Version of Zariski's Main Theorem for Polynomial Identity Rings

Matrix rings over polynomial identity rings. II

Matrix rings over polynomial identity rings. II

On Classical Quotients of Polynomial Identity Rings with Involution

On Classical Quotients of Polynomial Identity Rings with Involution

Structure of semiprime P.I. rings

In this paper we make an investigation into the structure of semiprime polynomial identity rings which is culminated by showing that each such ring R R has a unique maximal left quotient ring Q Q such that (1) Q Q is von Neumann regular with unity and (2) every regular element in R R is invertible in Q Q .

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.

Matrix rings over polynomial identity rings

We prove that if $A$ is an algebra over a field with at least $k$ elements, and $A$ satisfies ${x^k} = 0$, then ${A_n}$, the ring of $n$-by-$n$ matrices over $A$, satisfies ${x^q} = 0$, where $q = k{n^2} + 1$. Theorem 1.3 generalizes this result to rings: If $A$ is a ring satisfying ${x^k} = 0$, then for all $n$, there exists $q$ such that ${A_n}$ satisfies ${x^q} = 0$. Definitions. A