Equationally Compact Artinian Rings

Abstract

By a Noetherian (Artinian) ring=(R;+ , —, 0, ·) we mean an associative ring satisfying the ascending (descending) chain condition on left ideals. An arbitrary ringis said to beequationally compactif every system of ring polynomial equations with constants inis simultaneously solvable inprovided every finite subset is. (The reader is referred to [2; 8; 13; 14] for terminology and relevant results on equational compactness, and to [4] for unreferenced ring-theoretical results.) In this report a characterization of equationally compact Artinian rings is given - roughly speaking, these are the finite direct sums of finite rings and Prüfer groups; as consequences it is shown that an equationally compact ring satisfying both chain conditions is always finite, as is any Artinian ring which is a compact topological ring.