On semiperfect and perfect rings

Koh [2] has shown that a commutative ring with identity is semiperfect if and only if every simple right module has a projective cover. The purpose of this note is to show that the noncommutative version of Koh's characterization holds and to characterize right perfect rings as those for which every semisimple right module has a projective cover. All rings considered have identity. A submodule A of a right Rmodule M, R a ring, is small in M if A +B = M, for a submodule B of M implies B = M. The radical of a module MR, denoted J(M), is the intersection of all maximal submodules of M if there are any and is M otherwise, see e.g. [3 ]. The following proposition lists,some useful and known results.