Pre-self-injective duo rings

In this paper duo rings (every one-sided ideal is two sided) which are pre-self-injective (every proper homomorphic image is self-injective) are characterized. It will be shown that, for duo rings, a ring R is pre-self-injective if and only if it is one of the following rings: (1) a direct sum of a finite number of rank 0 maximal valuation rings; (2) a local ring whose maximal ideal M has composition length 2 and ${M^2} = 0$; or (3) a domain R in which every proper ideal is contained in only finitely many maximal ideals and which has, for each maximal ideal M and ideal $A \subseteq M,{R_M}/{A_M}$ being a rank 0 duo maximal valuation ring for the localization at M. The object of the paper is to extend the results of Klatt and Levy for commutative pre-self-injective rings [6]. Klatt and Levy’s characterization has already been extended [7] to self-injective rings for which all the proper cyclic modules are quasi-injective (called qc-rings). The qc-rings are direct sums of semisimple Artinian rings and duo rings. Jain, Singh, and Symonds have also studied the noncommutative problem by looking at rings with all the cyclic modules, which are not isomorphic to the ring, being quasi-injective (called PCQI-rings). The rings in the present paper do have the property that they are PCQI-rings. Hence, use can be made of the results in [5]. However, Jain, Singh, and Symonds left unresolved how to characterize prime PCQI-rings.