The most descriptive way to represent a finite Abelian group is as a direct product of cyclic groups. In this form, many important characteristics such as order, exponent, and rank are immediately evident. For instance, consider the group G of units of the ring of integers modulo 180. With this description of G, very little about its structure is readily apparent. On the other hand, if we are told that G is isomorphic to Z12 X Z2 X Z2 (see [9] or [11, p. 46]), then we may observe that G has order 48, exponent 12, and rank 3. More generally, one could ask for the structure of the group of units of any commutative ring R. When R is finite, we know from the Fundamental Theorem of Finite Abelian Groups the beautiful fact that the group of units of R, U(R), is isomorphic to a direct product of cyclic groups. So, it is natural to look for a method to express the group of units of any finite commutative ring as a direct product of cyclic groups. This problem has not been solved in general, but the solution to the special case in which R is a finite field is well known. In this case, U(R) is cyclic [8, p. 405]. Other familiar finite rings are those of the form R = F[x]/h(x)>, where F is a finite field and h(x) is a nonconstant polynomial in F[x]. For example, consider R = Z7[X]/((X + 4)8(X2 + 4)(X3 + 3)2). What is U(R) in this case? It turns out that a nice blend of elementary ring theory and group theory gives an algorithm which yields the answer: