Summary: Computations in Modules over Commutative Domains
Alkiviadis G. Akritas
and Gennadi I. Malaschonok
Summer 2002, ACA'02, Volos, Greece
This paper is a review of results on computational methods of linear algebra over
commutative domains. Methods for the following problems are examined: solution of
systems of linear equations, computation of determinants, computation of adjoint and
inverse matrices, computation of the characteristic polynomial of a matrix.
Let R be a commutative domain with identity, and K the field of quotients of R. This
paper reviews effective matrix methods in the domain R for the solution of standard
linear algebra problems. These problems are: solving linear systems in K, computing
the adjoint and inverse matrix, computing the matrix determinant and computing the
characteristic polynomial of a matrix.
The standard used to tackle these problems in a commutative domain R consists of
using the field of fractions K of this domain. The ring R may be canonically immersed
in the field K. To solve a problem in the commutative domain any algorithm that is
applicable over the field of fractions of this domain can be applied.
Unfortunately this way results in algorithms with suitable complexity only in the case