In this paper we canonically
represent the isomorphism classes of all rings whose additive group is a direct sum of
two infinite cyclic groups by a system of 4 by 2 matrices whose elements are
rational integers. It is then shown how the canonical forms can be used to solve
other problems relating to these rings. The results obtained are (1) that any
integral domain in this class of rings is isomorphic to a quadratic extension
of a subring of the integers, (2) the complete survey of rings in the class
under study which are decomposable as a direct sum, and (3) the complete
survey of rings in this class which are decomposable as an ordered product
which is not a direct sum. The paper concludes with a description of other
problems which can be solved by means of the canonical matrices using routine
calculations.
