Lee Klingler

The dynamic feedback cyclization problem for principal ideal domains

Abstract We prove that every controllable system over a principal ideal domain feeds back to a cyclic vector after a one-augmentation of the given system.

Representation type of commutative Noetherian rings. III. Global wildness and tameness

Introduction Preliminaries Dedekind-like rings Wildness Structure of a genus Substitute for conductor squares Isomorphism classes in a genus, idele group action Web of class groups Direct sums Finite normalization Appendix A Appendix B Bibliography.

Direct-sum decompositions over one-dimensional Cohen-Macaulay local rings

1 Alberto Facchini, Dipartimento di Matematica Pura e Applicata, Universita di Padova, Via Belzoni 7, I-35131 Padova, Italy, facchini@math.unipd.it 2 Wolfgang Hassler, Institut fur Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universitat Graz, Heinrichstrase 36/IV, A-8010 Graz, Austria, wolfgang.hassler@uni-graz.at 3 Lee Klingler, Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, FL 33431-6498, klingler@fau.edu 4 Roger Wiegand, Department of Mathematics, University of Nebraska, Lincoln, NE 68588-0323, rwiegand@math.unl.edu

Dynamic feedback over commutative rings

Abstract We call a commutative ring R a CA- α ( n ) ring if, for each n -dimensional reachable system ( F , G ) over R , the system augmented by rank α ( n ) projective modules, with F augmented by a zero map and G augmented by an identity map, is coefficient assignable. We show that, if R is a Dedekind domain, then R is a CA-( n − 1) ring. In particular, a principal ideal domain is a CA-( n − 1) ring . We also show that, if R is a ring with the GCS-property and the 2-generator property, then R is a CA-(2 n − 2) ring .

On feedback invariants for linear dynamical systems

Abstract The goal of this paper is to prove that, if R is a commutative ring containing a (non-zero) finitely generated maximal ideal M containing its annihilator, such that every unit of R/ M lifts to a unit of R , then the category of reachable systems over R is “wild” in the sense of classical representation theory. From this it follows that a canonical form for a reachable system over a PID is not likely to be found. More specifically, canonical forms are unlikely to be found for systems over Z and k[T] , when k is a field.

Integral representations of groups of square-free order☆

Abstract We describe the structure of the integral group ring Z G , when G has square-free order, as a subdirect sum of hereditary orders in skew group algebras. From this we deduce the structure of all genera of Z G -lattices. Our principal applications are the following (for groups G of square-free order). (i) We determine those G whose Z G -lattices satisfy uniqueness of the number of indecomposable summands. (ii) We determine those G whose Z G -lattices are direct sums of left ideals, (iii) For those G whose Z G -lattices are not direct sums of left ideals, we show that indecomposable Z G -lattices can be much larger than the ring Z G itself, despite the fact that Z G is of finite representation type and, over the p -adic completions of Z G , lattices always become direct sums of left ideals. (iv) We show that the ring structure of Q G determines the group G up to isomorphism.

Sweeping-similarity of matrices

Abstract This partly expository paper deals with a canonical-form problem for finite sets of matrices. The problem generalizes matrix equivalence, matrix similarity, and simultaneous equivalence of pairs of matrices [( A , B ) → ( PAQ , PBQ )], as well as some more complicated matrix problems that originated in work of Nazarova and Roiter.

C[y] is a CA-ring and coefficient assignment is properly weaker than feedback cyclization over a PID

Abstract Based on an “almost”-control-canonical form for reachable systems over PIDs, it is shown that coefficient assignment is possible over K [ y ], where K is an algebraically closed field of arbitrary characteristic. As consequences, the ring of polynomials C [ y ] over the complex numbers is a CA-ring, and there are (PID) CA-rings that are not FC-rings.

The Ordered Group of Invertible Ideals of a Prüfer Domain of Finite Character

Abstract Let D be a Prüfer domain, and denote by ± bℑ(D) the multiplicative group of all invertible fractional ideals of D, ordered by A ≤ B if and only if A ⊇ B. Denote by G i the value group of the valuation associated with the valuation ring D M i , where {M i } i∈I is the collection of all maximal ideals of D. In this note we prove that the natural map from ± bℑ(D) into ± b∏ i∈I G i is an isomorphism onto the cardinal sum ± b∐ i∈I G i if and only if D is h-local. As a corollary, the group of divisibility of an h-local Bézout domain is isomorphic to ± b∐ i∈I G i , the notation being as above.

Indecomposable modules of large rank over Cohen-Macaulay local rings

A commutative Noetherian local ring (R, m, k) is called Dedekind-like provided R is one-dimensional and reduced, the integral closure R is generated by at most 2 elements as an R-module, and m is the Jacobson radical of R. If M is an indecomposable finitely generated module over a Dedekind-like ring R, and if P is a minimal prime ideal of R, it follows from a classification theorem due to L. Klingler and L. Levy that M p must be free of rank 0, 1 or 2. Now suppose (R, m, k) is a one-dimensional Cohen-Macaulay local ring that is not Dedekind-like, and let P 1 ,... P t be the minimal prime ideals of R. The main theorem in the paper asserts that, for each non-zero t-tuple (n 1 ,... n t ) of non-negative integers, there is an infinite family of pairwise non-isomorphic indecomposable finitely generated R-modules M satisfying MP i ≅ (Rp i ) (n i ) for each i.