Rings over which cyclic modules are almost self-injective

A well known result by Osofsky (1964) states that if over a ring R every cyclic right module is injective, then R is semi-simple artinian. This has motivated the study of rings over which certain classes of cyclic modules satisfy a condition that is a generalization of injectivity, notable among them is a result by Koehler (1974), a characterization of rings over which all cyclic right modules are quasi-injective. In the present paper, right noetherian rings over which cyclic right modules are almost self-injective, are studied. The class of such rings include local serial rings, serial rings S with J(S)(2) = 0 and matrix rings [D M 0 S], where D is a local noetherian serial domain, S is an indecomposable serial ring with J(S)(2) = 0, and M-D(S) is a bimodule such that M-D is a torsion-free divisible module, MS is simple, the endomorphism ring End(MS) is the classical quotient ring of D, and if e(1), e(2),...., e(u) form a maximal orthogonal set of non-isomorphic indecomposable idempotents in S, then they can be so arranged that Me-1 not equal 0, for J = J(S), i < u, e(i)Je(i+1) not equal 0 and e(u)S is a minimal right ideal.