Systems of linear congruences
Canadian Journal of Mathematics - Journal Canadien de Mathematiques
Introduction. On recent occasions papers have been presented concerned with the problem of solving a system of linear congruences. Apparently the authors were not aware that this problem was solved very neatly and completely a long time ago by H. J. S. Smith (5; 6). One reason for this situation is that recent texts in the theory of numbers go only as far in the discussion of systems of congruences as one can with the most elementary tools ; whereas older texts, such as the one by Stieltjes (8"
... ne by Stieltjes (8" devote so much space to the discussion of the requisite matric theory that the reader is liable to lose sight of the elegant results concerning systems of congruences. Perhaps the time has come to give a new exposition of this material, particularly since this can be done in rather short compass to an audience whose background may be assumed to include acquaintance with the invariant factors and the Smith normal form of a matrix with elements in a principal ideal ring, $. In the final part of this paper we present some original work extending the discussion of systems of linear equations, and systems of linear congruences modulo an ideal, from the classical case over the rational domain to the case where the systems are over a set of integral elements, with a ^-basis, belonging to an associative algebra. Here we assume knowledge of the Hermite normal form of a matrix with elements in a principal ideal ring.