Virtually regular modules

In this paper, we call a right module [Formula: see text] (strongly) virtually regular if every (finitely generated) cyclic submodule of [Formula: see text] is isomorphic to a direct summand of [Formula: see text]. [Formula: see text] is said to be completely virtually regular if every submodule of [Formula: see text] is virtually regular. In this paper, characterizations and some closure properties of the aforementioned modules are given. Several structure results are obtained over commutative rings. In particular, the structures of finitely presented (strongly) virtually regular modules and completely virtually regular modules are fully determined over valuation domains. Namely, for a valuation domain [Formula: see text] with the unique nonzero maximal ideal [Formula: see text], we show that finitely presented (strongly) virtually regular modules are free if and only if [Formula: see text] is not principal; and that [Formula: see text] is principal if and only if finitely presented virtually regular modules are of the form [Formula: see text] for nonnegative integers [Formula: see text] Similarly, we prove that [Formula: see text] is principal if and only if finitely presented strongly virtually regular modules are of the form [Formula: see text], where [Formula: see text] are nonnegative integers. We also obtain that, [Formula: see text] admits a nonzero finitely presented completely virtually regular module [Formula: see text] if and only if [Formula: see text] is principal. Moreover, for a finitely presented [Formula: see text]-module [Formula: see text], we prove that: [Formula: see text] if [Formula: see text] is not a DVR, then [Formula: see text] is completely virtually regular if and only if [Formula: see text]; and [Formula: see text] if [Formula: see text] is a DVR, then [Formula: see text] is completely virtually regular if and only if [Formula: see text] Finally, we obtain a characterization of finitely generated virtually regular modules over the ring of integers.