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SOME CARDINALITY CONDITIONS FOR RING RADICALS

Abstract Motivated by some results and concepts from radical theory for abelian groups, we investigate the radical classes R of associative rings for which there is an infinite cardinal number α such that for every ring A and every a e R(A) we have a e R(B) for (i) some ideal B of A with |B| < α, (ii) some accessible subring B of A with |B| < α Only the class {0} satisfies (i), while the classes other than {0} which satisfy (ii) are the classes Dp of zerorings on divisible P-groups for sets P of primes. We also consider an analogous question for strict radical classes and subrings. Here the answer is much more complicated, and in fact undecidable in ZFC.