On the Compactness of the Structure Space of a Ring

If A is a ring, we denote by S = S(j4) the set of all primitive ideals of A. A topology may be defined on S in the following way: For 3Q§>, the closure of 3 in S is the set of all P& such that P3fl3. With this topology, S is the structure space [4] of A. It is well known that if A has a unit element, then S is compact [4; 5, p. 208]. Moreover, M. Schreiber [7] has recently observed that if every (two-sided) ideal of A is finitely generated, then S is again compact. However, since the condition that A have a unit element neither implies nor is implied by the condition that every ideal of A be finitely generated, it is clear that neither of these conditions is necessary in order that S be compact. In Theorem 2 of the present note we present a condition that is both necessary and sufficient for the compactness of S. The sufficient conditions already cited are both immediate consequences of our result. Although a direct proof of Theorem 2 can easily be formulated, we have chosen to obtain Theorem 2 as a consequence of a general lattice-theoretic result which we state as Theorem l.1