On the Restricted Minimum Condition for Rings

Generalizing Artinian rings, a ring R is said to have right restricted minimum condition ( r . RMC , for short) if R / A is an Artinian right R -module for any essential right ideal A of R . It is asked in Jain et al. [Cyclic Modules and the Structure of Rings, Oxford University Press, Oxford, 2012, 3.17 Questions (2)] that (i) Is a left self-injective ring with r . RMC quasi-Frobenius? (ii) Whether a serial ring with r . RMC must be Noetherian? We carry out a study of rings with r . RMC and determine when a right extending ring has r . RMC in terms of rings S M 0 R such that S is right Artinian, M Q is semisimple ( Q = Q ( R ) ) and R is a semiprime ring with Krull dimension 1. We proved that a left self-injective ring R with r . RMC is quasi-Frobenius if and only if Z r ( R ) = Z l ( R ) if and only if Z r ( R ) is a finitely generated left ideal and N ( R ) ∩ Soc ( R R ) is a finitely generated right ideal. Right serial rings with r . RMC are studied and proved that a non-singular serial ring has r . RMC if and only if it is a left Noetherian ring. Examples are presented to describe our results and to show that RMC is not symmetric for a ring.