ON A CLASSIFICATION OF PRIME RINGS

Abstract

Let m be a positive cardinal. We denote by P_r(m) (resp. P _t(m)) the class of all rings R for which m is the least cardinal such that all nonzero elements of R possess right (resp. left) insulators of cardinality less than m + 1. We also set P _r(m) = U_n ≤ m P_r(n). The classes Pr(m),> m 0, partition the class of all prime rings. Various descriptions of these classes are obtained. In particular if m is regular then P _r(m) contains just those rings R such that t(R) = 0 for all proper torsion preradicals t on Mod -R whose torsion classes are closed under direct products of fewer than m modules. Examples are provided which show that P _r(m) is non-empty for all m > 0 and which partially answer the question: for which cardinals m, n is P _r(m) ∩ P _t(n) nonempty?