On pairs of subrings with a common set of proper ideals

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 noncommutative setting. Consider the ring R= HomD(V ,V), where V is a vector space over a division ring D with dimD(V) = אω0 (ω0 is the first limit ordinal). Let K be any subfield of the center of R, M = { f ∈ HomD(V ,V) | dim f (V) < אω0}, and S= K +M. Then S and R have the same set of countably many prime ideals. Further, R and S have the same set of ideals since all of their proper ideals are prime ideals. (See Blair and Tsutsui [4]). We will first observe that examples of this nature are limited in the commutative setting. It is an immediate consequence of Theorem 1 that the only possible pairs of a commutative ring R and its proper unital subring S with the same set of proper ideals are fields. We will then briefly investigate pairs of subrings with a common ideal. This investigation will yield the fact that a pair of rings has the same set of prime ideals if and only if they have the same set of maximal ideals. Among other things, we consider properties that pass through a pair of rings with a common set of proper ideals. Hereafter, we reserve the term subring for a unital subring. Thus, not only a subring inherits its binary operations from its overring, but also it has the same identity element. We call subrings R and S of a ring (right-) ideally equal if they have the same set of proper (right) ideals, that is, I is a proper (right) ideal of R if and only if I is a proper (right) ideal of S. Two rings R and S are called P(M-)ideally equal if they have the same set of prime (maximal) ideals.