There are algorithms to solve all the problems addressed in this article over the integers. In other words, linear algebra is effective over the integers. See Linear Diophantine system for details.
The same solution applies to the same problems over a principal ideal domain, with the following modifications.
The notion of unimodular matrix of integers must be extended by calling unimodular a matrix over a integral domain whose determinant is a unit. This means that the determinant is invertible and implies that unimodular matrices are the invertible matrices such all entries of the inverse matrix belong to the domain.
To have an algorithmic solution of linear systems, a solution for a single linear equation in two unknowns is clearly required. In the case of the integers, such a solution is provided by extended Euclidean algorithm. Thus one needs that, for the considered principal ideal domain, there is an algorithm with a similar specification as the extended Euclidean algorithm. That is, given a and b in the principal ideal domain, there is an algorithm computing a unimodular matrix
Having such an algorithm, the Smith normal form of a matrix may be computed exactly as in the integer case, and this suffices to apply the method of Linear Diophantine system.
The main case where this is commonly used is the case of linear systems over the ring of univariate polynomials over a field. In this case, the extended Euclidean algorithm may be used. See polynomial greatest common divisor#Bézout’s identity and extended GCD algorithm for details.
~ source : http://en.wikipedia.org/wiki/Linear_equation_over_a_ring
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