On Virtually Regular Modules And Rings

A right module M over a ring R is said to be (strongly) virtually regular if every cyclic (finitely generated) submodule of M is isomorphic to a direct summand of M, and M is said to be completely virtually regular if every submodule of M is virtually regular. It is obtained that R is a right virtually regular ring if and only if every principal right ideal of R is projective, equivalently if and only if the ring R is right p.p. In our other principal structural assertion that depend on the “up to isomorphism” viewpoint, we describe semiperfect right strongly virtually regular rings as finite products of full matrix rings over right chain domains. We fully characterise the virtually regular abelian groups. Characterizing the completely virtually regular groups reduces to the case of torsion-free groups, and a torsion-free group is completely virtually regular if and only if it is free-reduced. When a torsion-free group has countable rank, or is a B 2 -group, then it is free-reduced if and only if it is free. We also show that the existence of a group of rank strictly less than the continuum that is free-reduced but not free is logically equivalent to the continuum hypothesis, and hence independent of ZFC.