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Let $R$ be a (not necessarily commutative) ring whose additive group is finitely generated and let $U_n(R) \subset GL_n(R)$ be the group of upper-triangular unipotent matrices over $R$. We study how the homology groups of $U_n(R)$ vary with $n$ from the point of view of representation stability. Our main theorem asserts that if for each $n$ we have representations $M_n$ of $U_n(R)$ over a ring $\mathbf{k}$ that are appropriately compatible and satisfy suitable finiteness hypotheses, then the rule $[n] \mapsto \widetilde{H}_i(U_n(R),M_n)$ defines a finitely generated OI-module. As a consequence, if $\mathbf{k}$ is a field then $dim \widetilde{H}_i(U_n(R),\mathbf{k})$ is eventually equal to a polynomial in $n$. We also prove similar results for the Iwahori subgroups of $GL_n(\mathcal{O})$ for number rings $\mathcal{O}$., Comment: 33 pages; minor update; to appear in Algebra & Number Theory

Spectral synthesis deals with the description of translation invariant function spaces. It turns out that the basic building blocks of this description are the exponential monomials, which are built up from exponential functions and polynomial functions. The author collaborated with Laczkovich [Math. Proc. Cambridge Philos. Soc. 143 (2007), no. 1, 103--120] proved that spectral synthesis holds on an Abelian group if and only if the torsion free rank of the group is finite. The author [Aequationes Math. 70 (2005), no. 1-2, 122--130] showed that the torsion free rank of an Abelian group is strongly related to the properties of polynomial functions on the group. Here we show that spectral synthesis holds on an Abelian group if and only if the ring of polynomial functions on the group is Noetherian.

We study finiteness conditions on essential extensions of simple modules over the quantum plane, the quantised Weyl algebra and Noetherian down-up algebras. The results achieved improve the ones obtained by Carvalho et al. (Carvalho et al., Injective modules over down-up algebras, Glasgow Math. J. 52A (2010), 53–59) for down-up algebras.

Let $k$ be a field. We show that a finitely generated simple Goldie $k$-algebra of quadratic growth is noetherian and has Krull dimension 1. Thus a simple algebra of quadratic growth is left noetherian if and only if it is right noetherian. As a special case, we see that if A is a finitely generated simple domain of quadratic growth then A is noetherian and by a result of Stafford every right and left ideal is generated by at most two elements. We conclude by posing questions and giving examples in which we consider what happens when the hypotheses are relaxed., 6 pages; fixed reference and corrected a misquoted statement from the literature

Let $\Lambda$ be a quasi $k$-Gorenstein ring. For each $d$th syzygy module $M$ in mod $\Lambda$ (where $0 \leq d \leq k-1$), we obtain an exact sequence $0 \to B \to M \bigoplus P \to C \to 0$ in mod $\Lambda$ with the properties that it is dual exact, $P$ is projective, $C$ is a $(d+1)$st syzygy module, $B$ is a $d$th syzygy of Ext$_{\Lambda}^{d+1}(D(M), \Lambda)$ and the right projective dimension of $B^*$ is less than or equal to $d-1$. We then give some applications of such an exact sequence as follows. (1) We obtain a chain of epimorphisms concerning $M$, and by dualizing it we then get the spherical filtration of Auslander and Bridger for $M^*$. (2) We get Auslander and Bridger's Approximation Theorem for each reflexive module in mod $\Lambda ^{op}$. (3) We show that for any $0 \leq d \leq k-1$ each $d$th syzygy module in mod $\Lambda$ has an Evans-Griffith representation. As an immediate consequence of (3), we have that, if $\Lambda$ is a commutative noetherian ring with finite self-injective dimension, then for any non-negative integer $d$, each $d$th syzygy module in mod $\Lambda$ has an Evans-Griffith representation, which generalizes an Evans and Griffith's result to much more general setting., Comment: 13 pages

Let F be a field, and let R be a finitely-generated F-algebra, which is a domain with quadratic growth. It is shown that either the center of R is a finitely-generated F-algebra or R satisfies a polynomial identity (is PI) or else R is algebraic over F. Let r ∈ R be not algebraic over F and let C be the centralizer of r. It is shown that either the quotient ring of C is a finitely-generated division algebra of Gelfand-Kirillov dimension 1 or R is PI.

One of the major open problems in noncommutative algebraic geometry is the classification of noncommutative surfaces, and this paper resolves a significant case of this problem. Specifically, let S denote the 3-dimensional Sklyanin algebra over an algebraically closed field k and assume that S is not a finite module over its centre. (This algebra corresponds to a generic noncommutative P^2.) Let A be any connected graded k-algebra that is contained in and has the same quotient ring as a Veronese ring S^(3n). Then we give a reasonably complete description of the structure of A. This is most satisfactory when A is a maximal order, in which case we prove, subject to a minor technical condition, that A is a noncommutative blowup of S^(3n) at a (possibly non-effective) divisor on the associated elliptic curve E. It follows that A has surprisingly pleasant properties; for example it is automatically noetherian, indeed strongly noetherian, and has a dualizing complex., 55 pages

The aim of the present paper is to show that, under some conditions, the uniform dimension of a ring R is the same as the uniform dimension of a skew Poincaré-Birkhoff-Witt extension built on R. El propósito de este artículo es mostrar que bajo ciertas condiciones, la dimensión uniforme de un anillo R coincide con la dimensión uniforme de una extensión Poincaré-Birkhoff-Witt torcida de R.

We study prime ideals in skew power series rings T:= R[[y; τ, δ]], for suitably conditioned complete right Noetherian rings R, automorphisms τ of R, and τ-derivations δ of R. Such rings were introduced by Venjakob, motivated by issues in noncommutative Iwasawa theory. Our main results concern "Cutting Down" and "Lying Over." In particular, assuming that τ extends to a compatible automorphsim of T, we prove: If I is an ideal of R, then there exists a τ-prime ideal P of T contracting to I if and only if I is a τ-δ-prime ideal of R. Consequently, under the more specialized assumption that δ = τ - id (a basic feature of the Iwasawa-theoretic context), we can conclude: If I is an ideal of R, then there exists a prime ideal P of T contracting to I if and only if I is a τ-prime ideal of R. Our approach depends essentially on two key ingredients: First, the algebras considered are Zariskian (in the sense of Li and Van Oystaeyen), and so the ideals are all topologically closed. Second, topological arguments can be used to apply previous results of Goodearl and the author on skew polynomial rings. © 2012 Hebrew University Magnes Press.

We prove that for a noetherian semilocal ring $R$ with exactly $k$ isomorphism classes of simple right modules the monoid $V^*(R)$ of isomorphism classes of countably generated projective right (left) modules, viewed as a submonoid of $V^*(R/J(R))$, is isomorphic to the monoid of solutions in $(\No \cup\{\infty\})^k$ of a system consisting of congruences and diophantine linear equations. The converse also holds, that is, if $M$ is a submonoid of $(\No \cup\{\infty\})^k$ containing an order unit $(n_1,..., n_k)$ of $\No^k$ which is the set of solutions of a system of congruences and linear diophantine equations then it can be realized as $V^*(R)$ for a noetherian semilocal ring such that $R/J(R)\cong M_{n_1}(D_1)\times ... \times M_{n_k}(D_k)$ for suitable division rings $D_1,..., D_k$., 41 pages

We study prime ideals in skew power series rings $T:=R[[y;\tau,\delta]]$, for suitably conditioned right noetherian complete semilocal rings $R$, automorphisms $\tau$ of $R$, and $\tau$-derivations $\delta$ of $R$. These rings were introduced by Venjakob, motivated by issues in noncommutative Iwasawa theory. Our main results concern "Cutting Down" and "Lying Over." In particular, under the additional assumption that $\delta = \tau - id$ (a basic feature of the Iwasawa-theoretic context), we prove: If $I$ is an ideal of $R$, then there exists a prime ideal $P$ of $S$ contracting to $I$ if and only if $I$ is a $\delta$-stable $\tau$-prime ideal of $R$. Our approach essentially depends on two key ingredients: First, the algebras considered are zariskian (in the sense of Li and Van Oystaeyen), and so the ideals are all topologically closed. Second, topological arguments can be used to apply previous results of Goodearl and the author on skew polynomial rings., Comment: Revised. 13 pages

This paper constitutes the first part of a program to classify all affine prime regular Hopf algebras $H$ of Gelfand-Kirillov dimension one over an algebraically closed field of characteristic zero. We prove a number of properties of such an algebra, list some classes of examples, and then prove that - when the PI-degree of $H$ is prime - our list contains all such algebras., Comment: 43 pages

The definition of p-injectivity motivates us to generalize the notion of injectivity and projectivity, noted respectively C-injectivity and C-projectivity. Noetherian, semi-simple Artinian, quasi-Frobenius, regular hereditary and self-injective regular rings are considered in terms of C-injectivity and C-projectivity. Partial answers are given to Matlis’ Problem and Boyle’s Conjecture.

We study the "q-commutative" power series ring R:=k_q[[x_1,...,x_n]], defined by the relations x_ix_j = q_{ij}x_j x_i, for multiplicatively antisymmetric scalars q_{ij} in a field k. Our results provide a detailed account of prime ideal structure for a class of noncommutative, complete, local, noetherian domains having arbitrarily high (but finite) Krull, global, and classical Krull dimension. In particular, we prove that the prime spectrum of R is normally separated and is finitely stratified by commutative noetherian spectra. Combining this normal separation with results of Chan, Wu, Yekutieli, and Zhang, we are able to conclude that R is catenary. Following the approach of Brown and Goodearl, we also show that links between prime ideals are provided by canonical automorphisms. Moreover, for sufficiently generic q_{ij}, we find that R has only finitely many prime ideals and is a UFD (in the sense of Chatters)., Revised; to appear in Algebras and Representation Theory. In this (final) version, catenarity is established, by combining normal separation with results of Chan, Wu, Yekutieli, and Zhang.

For k a field of arbitrary characteristic, and R a k-algebra, we show that the PI degree of an iterated skew polynomial ring R[x_1;\tau_1,\delta_1]...b[x_n;\tau_n,\delta_n] agrees with the PI degree of R[x_1;\tau_1]...b[x_n;\tau_n] when each (\tau_i,\delta_i) satisfies a q_i-skew relation for q_i \in k^{\times} and extends to a higher q_i-skew \tau_i-derivation. We confirm the quantum Gel'fand-Kirillov conjecture for various quantized coordinate rings, and calculate their PI degrees. We extend these results to completely prime factor algebras., Comment: 43 pages; Submitted to the Journal of Algebra

This is a survey of the known properties of Iwasawa algebras, which are completed group rings of compact p-adic analytic groups with coefficients the ring Zp of p-adic integers or the field Fp of p elements. A number of open questions are also stated., 27 pages, submitted to the John Coates volume of Documenta Mathematica

Let B(X,L,s) be the twisted homogeneous coordinate ring of an irreducible variety X over an algebraically closed field k with dim X > 1. Assume that c in X and s in Aut(X) are in sufficiently general position. We show that if one follows the commutative prescription for blowing up X at c, but in this noncommutative setting, one obtains a noncommutative ring R=R(X,c,L,s) with surprising properties. In particular: (1) R is always noetherian but never strongly noetherian. (2) If R is generated in degree one then the images of the R-point modules in qgr(R) are naturally in (1-1) correspondence with the closed points of X. However, both in qgr(R) and in gr(R), the R-point modules are not parametrized by a projective scheme. (3) qgr R has finite cohomological dimension yet H^1(R) is infinite dimensional. This gives a more geometric approach to results of the second author who proved similar results for X=P^n by algebraic methods., Latex, 42 pages

We consider algebras over a field K defined by a presentation K , where $R$ consists of n choose 2 square-free relations of the form x_i x_j = x_k x_l with every monomial x_i x_j, i different from j, appearing in one of the relations. Certain sufficient conditions for the algebra to be noetherian and PI are determined. For this, we prove more generally that right noetherian algebras of finite Gelfand-Kirillov dimension defined by homogeneous relations satisfy a polynomial identity. The structure of the underlying monoid, defined by the same presentation, is described. This is used to derive information on the prime radical and minimal prime ideals. Some examples are described in detail. Earlier, Etingof, Schedler and Soloviev, Gateva-Ivanova and Van den Bergh, and the authors considered special classes of such algebras in the contexts of noetherian algebras, Grobner bases, finitely generated solvable groups, semigroup algebras, and set theoretic solutions of the Yang-Baxter equation., Comment: Work supported in part by Onderzoeksraad of Vrije Universiteit Brussel, Fonds voor Wetenschappelijk Onderzoek (Belgium), and KBN research grant 2P03A 030 18 (Poland)

We develop a method of reducing the size of quantum minors in the algebra of n x n quantum matrices. The method is used to show that quantum determinantal factor rings of n x n quantum matrices over the complex numbers are maximal orders, when the parameter q is transcendental over the rational numbers., Comment: 16 pages

The prime and primitive spectra of Oq(fn), the multiparameter quantized coordinate ring of affine n-space over an algebraically closed field k, are shown to be topological quotients of the corresponding classical spectra, specO(fn) and maxo(fn) ≈kn, provided the multiplicative group generated by the entries of q avoids -1. ©2000 American Mathematical Society.

Differential polynomial rings over a universal field and localized twisted polynomial rings over a separably closed field of non-zero characteristic twisted by the Frobenius endomorphism were the first domains not divisions rings that were shown to have every simple module injective (see [C] and [C-J]). By modifying the separably closed condition for the polynomial rings twisted by the Frobenius, the conditions of every simple being injective and only a single isomorphism class of simple modules were shown to be independent (see [O]). In this paper we continue the investigation of injective cyclic modules over twisted polynomial rings with coefficients in a commutative field.

A condition on an affine central subalgebra Z of a noetherian algebra A of finite Gelfand–Kirillov dimension, which we call here unruffledness , is shown to be equivalent in some circumstances to the flatness of A as a Z -module. Unruffledness was studied by Borho and Joseph in work on enveloping algebras of complex semisimple Lie algebras, and we discuss applications of our result to enveloping algebras, as well as beginning the study of this condition for more general algebras.