RINGS IN WHICH MINIMAL LEFT IDEALS ARE PROJECTIVE

Let R be an associative ring with identity. Then the left socle of R is a direct summand of R as a right R-module if and only if it is projective as a left R-module and contains no infinite sets of orthogonal idempotents. This implies, for example, that a ring with finitely generated left socle and no nilpotent minimal left ideals is a ring direct sum of a semisimple artinian ring and a ring with zero left socle. © 1969 by Pacific Journal of Mathematics.