Factorizations of Elements in Noncommutative Rings: A Survey

We survey results on factorizations of non zero-divisors into atoms (irreducible elements) in noncommutative rings. The point of view in this survey is motivated by the commutative theory of non-unique factorizations. Topics covered include unique factorization up to order and similarity, 2-firs, and modular LCM domains, as well as UFRs and UFDs in the sense of Chatters and Jordan and generalizations thereof. We recall arithmetical invariants for the study of non-unique factorizations, and give transfer results for arithmetical invariants in matrix rings, rings of triangular matrices, and classical maximal orders as well as classical hereditary orders in central simple algebras over global fields.Comment: 50 pages, comments welcom

A Survey on Fixed Divisors

In this article, we compile the work done by various mathematicians on the topic of the fixed divisor of a polynomial. This article explains most of the results concisely and is intended to be an exhaustive survey. We present the results on fixed divisors in various algebraic settings as well as the applications of fixed divisors to various algebraic and number theoretic problems. The work is presented in an orderly fashion so as to start from the simplest case of $\Z,$ progressively leading up to the case of Dedekind domains. We also ask a few open questions according to their context, which may give impetus to the reader to work further in this direction. We describe various bounds for fixed divisors as well as the connection of fixed divisors with different notions in the ring of integer-valued polynomials. Finally, we suggest how the generalization of the ring of integer-valued polynomials in the case of the ring of $n \times n$ matrices over $\Z$ (or Dedekind domain) could lead to the generalization of fixed divisors in that setting.Comment: Accepted for publication in Confluentes Mathematic

Factoring the Adjoint and Maximal Cohen--Macaulay Modules over the Generic Determinant

A question of Bergman asks whether the adjoint of the generic square matrix over a field can be factored nontrivially as a product of square matrices. We show that such factorizations indeed exist over any coefficient ring when the matrix has even size. Establishing a correspondence between such factorizations and extensions of maximal Cohen--Macaulay modules over the generic determinant, we exhibit all factorizations where one of the factors has determinant equal to the generic determinant. The classification shows not only that the Cohen--Macaulay representation theory of the generic determinant is wild in the tame-wild dichotomy, but that it is quite wild: even in rank two, the isomorphism classes cannot be parametrized by a finite-dimensional variety over the coefficients. We further relate the factorization problem to the multiplicative structure of the \Ext--algebra of the two nontrivial rank-one maximal Cohen--Macaulay modules and determine it completely.Comment: 44 pages, final version of the work announced in math.RA/0408425, to appear in the American Journal of Mathematic

Complexity spectrum of some discrete dynamical systems

We first study birational mappings generated by the composition of the matrix inversion and of a permutation of the entries of $3 \times 3$ matrices. We introduce a semi-numerical analysis which enables to compute the Arnold complexities for all the $9!$ possible birational transformations. These complexities correspond to a spectrum of eighteen algebraic values. We then drastically generalize these results, replacing permutations of the entries by homogeneous polynomial transformations of the entries possibly depending on many parameters. Again it is shown that the associated birational, or even rational, transformations yield algebraic values for their complexities.Comment: 1 LaTex fil

Matrix factorizations and link homology

For each positive integer n the HOMFLY polynomial of links specializes to a one-variable polynomial that can be recovered from the representation theory of quantum sl(n). For each such n we build a doubly-graded homology theory of links with this polynomial as the Euler characteristic. The core of our construction utilizes the theory of matrix factorizations, which provide a linear algebra description of maximal Cohen-Macaulay modules on isolated hypersurface singularities.Comment: 108 pages, 61 figures, latex, ep