Semigroups presented by one relation and satisfying the Church-Rosser property

This paper is devoted to the study of semigroups presented by a single defining relationA=B and satisfying the Church-Rosser property. Translated fromMatematicheskie Zametki, Vol. 61, No. 1, pp. 114–118, January, 1997. Translated by A. I. Shtern     

Noetherian Equations for Matrices

Let L|K be a finite Galois extension. Using central simple algebras we deal with the crossed representations of G = Gal(L|K) over L which are defined as mappings X of G into GL_n(L) satisfying X = X X. The last equation is the Noetherian equation in case n=1. Furtheron, more general crossed projective representations are considered which obey an equation X X = Xf_, where f_, L.     

First-order rigidity of rings satisfying polynomial identities

We prove that every finitely generated Noetherian ring which satisfies a polynomial identity is first-order rigid. This generalizes a result of Aschenbrenner, Khélif, Naziazeno and Scanlon on commutative rings.     

von Neumann regular modules

In this paper, we introduce von Neumann regular modules and give many characterizations of von Neumann regular modules. Further, we investigate the relations between von Neumann regular modules and other classical modules. Finally, we characterize Noetherian von Neumann regular modules.     

Primary decomposition of modules: A computational differential approach

We study primary submodules and primary decompositions from a differential and computational point of view. Our main theoretical contribution is a general structure theory and a representation theorem for primary submodules of an arbitrary finitely generated module over a polynomial ring. We characterize primary submodules in terms of differential operators and punctual Quot schemes. Moreover, we introduce and implement an algorithm that computes a minimal differential primary decomposition for a module.     

Lazy bases: a minimalist constructive theory of Noetherian rings

We give a constructive treatment of the theory of Noetherian rings. We avoid the usual restriction to coherent rings; we can even deal with non‐discrete rings. We introduce the concept of rings with certifiable equality which covers discrete rings and much more. A ring R with certifiable equality can be fitted with a partial ideal membership test for ideals of R. Lazy bases of ideals of R [X ] are introduced in order to derive a partial ideal membership test for ideals of R [X ]. It is then proved that if R is Noetherian, then so is R [X ]. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Semiperfect Semidistributive Rings

Mathematical Notes -     

Naïve noncommutative blowups at zero-dimensional schemes

In an earlier paper [D.S. Keeler, D. Rogalski, J.T. Stafford, Naïve noncommutative blowing up, Duke Math. J. 126 (2005) 491–546, MR 2120116], we defined and investigated the properties of the naïve blowup of an integral projective scheme X at a single closed point. In this paper we extend those results to the case when one naïvely blows up X at any suitably generic zero-dimensional subscheme Z. The resulting algebra A has a number of curious properties; for example it is noetherian but never strongly noetherian and the point modules are never parametrized by a projective scheme. This is despite the fact that the category of torsion modules in qgr-A is equivalent to the category of torsion coherent sheaves over X. These results are used in the companion paper [D. Rogalski, J.T. Stafford, A class of noncommutative projective surfaces, in press] to prove that a large class of noncommutative surfaces can be written as naïve blowups.