Rings on the direct product of two cyclic groups

The focus of this paper is a classification of rings whose additive group is the direct product of two cyclic groups. Such rings are represented by a quotient ring of the polynomials with integer coefficients. The paper begins with an overview of general ring theory including the Chinese Remainder theorem and the theory of local/irreducible rings. We then introduce Hensel's lemma which is later used as the main tool for classifying rings on the direct product of two cyclic groups. It is shown that two of these rings are isomorphic if and only if there is a solution to a particular quadratic equation in two variables mod n. We derive a new form of Hensel's lemma that applies directly to quadratic equations in two variables. It is used to systematically solve the quadratics in question and thus obtain a complete classification of rings on the direct product of two cyclic groups.