Modules over a principal ideal domain

This thesis determines the structure of certain modules over a principal ideal domain, namely the divisible modules and the finitely generated modules. The author proves that any divisible module M over a principal ideal domain D is isomorphic to a direct sum of modules each of which is a copy of the quotient field KD or to Dp" for various primes p ? D. The author also proves that a finitely generated module is isomorphic to a finite product of cyclic modules. Any finitely generated module is characterized up to isomorphism by certain algebraic invariants. The results on finitely generated modules are then applied to vector space theory to develop the rational and Jordan canonical forms

Chain Complexes over Principal Ideal Domains

Chain complexes are studied here as an abstract algebraic generalisation of geometric simplicial complexes. From this point of view, we extend the notions of shellability and of a cone, which are both defined for simplicial complexes, to chain complexes. We define a cone for chain complexes in an abstract way abandoning the geometrical idea of an apex and compare it with mapping cones. Indeed, there are cones which cannot be regarded as mapping cones, in contrast to the simplicial case. And conversely, we name certain conditions on which a mapping cone is a cone matching our definition. Our notion of shellability given here for chain complexes is a generalisation of this well-known term which is defined for simplicial complexes as well as for regular finite CW-complexes. But in contrast to shellable simplicial complexes, there is no information about the homology of shellable chain complexes, so we claim additional conditions on them which imitate other properties of simplicial complexes. This leads to our notions of regular and totally regular chain complexes. We obtain complete homological information for totally regular chain complexes which have a specific augmentation map. In the end, we consider mapping cones over shellable or regular chain complexes and show that they also are shellable or regular, respectively

FINITELY GENERATED MODULES OVER A PRINCIPAL IDEAL DOMAIN

This thesis covers the main theory of modules: modules, submodules, cosets, quotient modules, homomorphisms, ideals, direct sums, and some related topics. Using these notions, a theorem on the structure of finitely generated modules over domains of principal ideals is proved. As an application of this theorem, the theory of the structure of normal forms of matrices over various fields is presented

Null ideals of matrices over residue class rings of principal ideal domains

Given a square matrix $A$ with entries in a commutative ring $S$, the ideal of $S[X]$ consisting of polynomials $f$ with $f(A) =0$ is called the null ideal of $A$. Very little is known about null ideals of matrices over general commutative rings. We compute a generating set of the null ideal of a matrix in case $S = D/dD$ is the residue class ring of a principal ideal domain $D$ modulo $d\in D$. We discuss two applications. At first, we compute a decomposition of the $S$-module $S[A]$ into cyclic $S$-modules and explain the strong relationship between this decomposition and the determined generating set of the null ideal of $A$. And finally, we give a rather explicit description of the ring \IntA of all integer-valued polynomials on $A$

Diagonalization of matrices over graded principal ideal domains

AbstractThe 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

Learning weighted automata over principal ideal domains

In this paper, we study active learning algorithms for weighted automata over a semiring. We show that a variant of Angluin’s seminal L⋆ algorithm works when the semiring is a principal ideal domain, but not for general semirings such as the natural numbers