Similarity of matrices over Artinian principal ideal rings

This paper discusses the theory of similarity of matrices over a commutative Artinian principal ideal ring R. It is shown that for the class matrices A such that R[A] is R-free a “rational” canonical form is available.     

Determinants of matrices over commutative finite principal ideal rings

In this paper, the determinants of $n\times n$ matrices over commutative finite chain rings and over commutative finite principal ideal rings are studied. The number of $n\times n$ matrices over a commutative finite chain ring R of a fixed determinant a is determined for all $a\in R$ and positive integers n. Using the fact that every commutative finite principal ideal ring is a product of commutative finite chain rings, the number of $n\times n$ matrices of a fixed determinant over a commutative finite principal ideal ring is shown to be multiplicative, and hence, it can be determined. These results generalize the case of matrices over the ring of integers modulo m.     

Commutators of trace zero matrices over principal ideal rings

We prove that for every trace zero square matrix A of size at least 3 over a principal ideal ring R, there exist trace zero matrices X, Y over R such that XY ? YX = A. Moreover, we show that X can be taken to be regular mod every maximal ideal of R. This strengthens our earlier result that A is a commutator of two matrices (not necessarily of trace zero), and in addition, the present proof is simpler than the earlier one.     

Lexicodes over finite principal ideal rings

Let R be a (possibly noncommutative) finite principal ideal ring. Via a total ordering of the ring elements and an ordered basis a lexicographic ordering of the module \(R^n\) is produced. This is used to set up a greedy algorithm that selects vectors for which all linear combinations with the previously selected vectors satisfy a pre-specified selection property and updates the to-be-constructed code to the linear hull of the vectors selected so far. The output is called a lexicode. This process was discussed earlier in the literature for fields and chain rings. In this paper we investigate the properties of such lexicodes over finite principal ideal rings and show that the total ordering of the ring elements has to respect containment of ideals for the algorithm to produce meaningful results. Only then it is guaranteed that the algorithm is exhaustive and thus produces codes that are maximal with respect to inclusion. It is further illustrated that the output of the algorithm heavily depends on the total ordering and chosen basis.     

On the factorization of polynomial matrices over the domain of principal ideals

We consider the problem of decomposition of polynomial matrices over the domain of principal ideals into a product of factors of lower degrees with given characteristic polynomials. We establish necessary and, under certain restrictions, sufficient conditions for the existence of the required factorization.     

MDS codes over finite principal ideal rings

The purpose of this paper is to study codes over finite principal ideal rings. To do this, we begin with codes over finite chain rings as a natural generalization of codes over Galois rings GR(p ^e , l) (including ). We give sufficient conditions on the existence of MDS codes over finite chain rings and on the existence of self-dual codes over finite chain rings. We also construct MDS self-dual codes over Galois rings GF(2 ^e , l) of length n = 2 ^l for any a ≥ 1 and l ≥ 2. Torsion codes over residue fields of finite chain rings are introduced, and some of their properties are derived. Finally, we describe MDS codes and self-dual codes over finite principal ideal rings by examining codes over their component chain rings, via a generalized Chinese remainder theorem.      

Diagonalization of matrices over graded principal ideal domains

The main result of this paper is the analogue of the classical diagonal reduction of matrices over PIDs, for graded principal ideal domains. A method for diagonalizing graded matrices over a graded principal ideal domain is obtained. In Section 2 we emphasis on some applications. A procedure is given to decide whether or not a matrix defined over an ordinary Dedekind domain (i.e. nongraded), with cyclic class group, is diagonalizable. In case the answer is positive the diagonal form can be calculated. This can be done by taking a suitable graded PID which has the Dedekind domain as its part of degree zero. It turns out that, even in the case where diagonalization of a matrix over the part of degree zero is not possible, the diagonal representation over the graded ring contains useful information. The main reason for this is that the graded ring hasn't essentially more units than its part of degree zero. We illustrate this by considering the problem of von Neumann regularity of a matrix over a Gr-PID and to matrices over Dedekind domains with cyclic class group. These problems were the original motivation for studying diagonalization over graded rings.     

Combinatorial coverings from geometries over principal ideal rings

A t-(v, k, λ) covering is an incidence structure with v points, each block incident on exactly k points, such that every set of t distinct points is incident on at least λ blocks. By considering certain geometries over finite principal ideal rings, we construct infinite families of t-(v, k, λ) coverings having many interesting combinatorial properties. © 1999 John & Sons, Inc. J Combin Designs 7: 247–268, 1999     

Dynamic feedback over principal ideal domains and quotient rings

Let R be a principal ideal domain. In this paper we prove that, for a large class of linear systems, dynamic feedback over R is equivalent to static feedback over a quotient ring of R. In particular, when R is the ring of integers Z one has that the static feedback classification problem over finite rings is equivalent to the dynamic feedback classification problem over Z restricted to a special type of system.     

Matrix graphs and MRD codes over finite principal ideal rings

Let R be a finite principal ideal ring and $m,n,d$ positive integers. In this paper, we study the matrix graph over R which is the graph whose vertices are $m\times n$ matrices over R and two matrices A and B are adjacent if and only if $0<\mathrm{rank}(A-B)<d$. We show that this graph is a connected vertex transitive graph. The distance, diameter, independence number, clique number and chromatic number of this graph are also determined. This graph can be applied to study MRD codes over R. We obtain that a maximal independent set of the matrix graph is a maximum rank distance (MRD) code and vice versa. Moreover, we show the existence of linear MRD codes over R.     

A principal ideal theorem analogue for modules over commutative rings

The Principal Ideal Theorem states that if Re is a commutative Noetherian ring and ? is a prime ideal of Re which is minimal over a principal ideal then Pe has height at most 1. Also, if Re is a (not necessarily Noetherian) UFD and Pe is a prime ideal of Re minimal over a principal ideal then Pe has height at most 1. We shall show that there are analogues for modules over commutative rings, but they hold only in special cases.     

Restricted left principal ideal rings

A ring is an LD-ring ifR is left bounded, ifR/J is a left Artinian left principal ideal ring for every proper idealJ inR, and ifR has finite left Goldie dimension. IfR is non-Artinian thenR is an order in a simple Artinian ringS. The ideal theory of LD-rings is investigated, and we discuss some conditions under which an LD-ring is an hereditary ring, and some under which an LD-ring is a Noetherian, bounded, maximal Asano order. A central localization of an LD-ring is an LD-ring, and the center of some LD-rings is a Krull-domain. This research was supported in part by the National Science Foundation Grant GP 23861.     

Homogeneous weights of matrix product codes over finite principal ideal rings

In this paper, the homogeneous weights of matrix product codes over finite principal ideal rings are studied and a lower bound for the minimum homogeneous weights of such matrix product codes is obtained.     

Homogeneous metric and matrix product codes over finite commutative principal ideal rings

In this paper, a necessary and sufficient condition for the homogeneous distance on an arbitrary finite commutative principal ideal ring to be a metric is obtained. We completely characterize the lower bound of homogeneous distances of matrix product codes over any finite principal ideal ring where the homogeneous distance is a metric. Furthermore, the minimum homogeneous distances of the duals of such codes are also explicitly investigated.