Key points are not available for this paper at this time.
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. We study the RMC for matrix extensions of a ring R and for the group-ring RG. Among other things, it is proven that having r.RMC is a Morita invariant property for rings. For n≥2, the upper triangular n by n matrix ring over R has r.RMC if and only if RR is Artinian. If G is a not torsion abelian group then RG has r.RMC if and only if R is a semisimple ring and G≃H⊕Z where H is a finite group whose order is invertible in R. If X is a completely regular topological space, then the ring C(X) has RMC if and only if X is finite. Many examples of rings with r.RMC are presented.
Z. et al. (Sat,) studied this question.