We consider a sequence defined by the number of positive solutions to a sequence of systems of Diophantine equations. We derive some bounds on the solutions to demonstrate that the terms of the sequence are finite. We develop an algorithm for 1 computing an arbitrary term of the sequence, and consider a graph-theoretic approach to computing the same.

Throughout, all rings are associative with an identity element but not necessarily commutative. During the decade of 1980s, a series of papers appeared in the Canadian Journal of Mathematics [1, 2] that had investigated pairs of commutative rings with the same set of prime ideals. In this paper, we consider some generalizations of that study in the… (More)