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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.

A system of canonical forms for rings on a direct sum of two infinite cyclic groups