**Title: **Rings with cyclic modules almost self-injective

**Speaker: **Surjeet Singh (Panjab University, Chandigarh, India)

**Abstract: **If a ring $R$ is such that every cyclic right $R$-module is injective, as proved by Osofsky in 1964, $R$ is semi-simple artinian. This has motivated others to find structure of rings $R$ over which certain class of modules have a well defined property $P$. For instance, Koehler and Ahsan independently studied rings over which cyclic right modules are quasi-injective, Faith studied rings over which proper cyclic right modules are injective. Baba had introduced the concept of almost relative injectivity in 1989. If a module $M$ is almost $M$-injective, then $M$ is said to be almost self-injective. Any quasi-injective module is almost self-injective.

A ring $R$ over which all cyclic right modules are almost self-injective, is called a right $cai$-ring. It is proved that a right noetherian ring $R$ that is a right $cai$, is a finite direct sum of local uniserial rings, serial ring with the square of its radical zero, and $2\times 2$ matrix ring where $D$ is a local, noetherian, serial domain, $S$ a serial ring with $J(S)^{2}$ = 0, $M$ a $(D,S)$-bimodule such that $M_{S}$ is simple, $_{D}M$ is a torsion-free divisible module and $End(M_{S})$ is the classical quotient ring of $D$, and certain other conditions.