L. Levy has characterised a
commutative noetherian ring in which every proper homomorphic image is
self-injective to be a Dedekind domain, or a principal ideal ring with descending
chain condition, or a local ring whose maximal ideal M has composition length 2 and
satisfies M2= 0. The object of this paper is to generalise Levy’s result to the
noncommutative case by studying right noetherian rings in which every proper
homomorphic image is a right q-ring. Since every commutative self-injective ring is a
q-ring, one can get Levy’s result as a special case of Theorems 2.12 and
2.13.
