Rings with restricted injective condition. (English) Zbl 0703.16003

The relationship and possible equality of the following classes are considered: (1) the rings whose singular right modules are injective (right SI-rings), (2) the rings whose cyclic right modules all have injective singular submodules (right CSI-rings), and (3) the rings whose singular cyclic right modules are all injective (right RIC-rings). Classes (1) and (3) have been studied extensively before; class (2) is introduced in this paper. A right RIC-ring R is right SI if and only if \(R/Soc(R_ R)\) has finite uniform dimension. A right RIC-ring R is right noetherian right SI if every right ideal of R is contained as an essential submodule in a direct summand of \(R_ R\). A von Neumann regular right CSI-ring is a left and right SI-ring. If R is a right CSI- ring such that \(R_ R\) has a cyclic injective hull, then R is a right artinian, right and left SI-ring. If R is a right CSI-ring with \(Soc(R_ R)=0\), then R does not contain an infinite direct sum of nonzero ideals. The authors conjecture that right CSI-rings are always SI-rings.
Reviewer: M.L.Teply


16D50 Injective modules, self-injective associative rings
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
16P40 Noetherian rings and modules (associative rings and algebras)
16S90 Torsion theories; radicals on module categories (associative algebraic aspects)
16E10 Homological dimension in associative algebras
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