Abstract
A ring R is said to have right restricted minimum condition ( for short) if R/A is an Artinian right R-module for any essential right ideal A of R. We study the 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 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 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 if and only if X is finite. Many examples of rings with r.RMC are presented.
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