Virtually regular modules

We call a right module $M$ (strongly) virtually regular if every (finitely generated) cyclic submodule is isomorphic to a direct summand. $M$ is said to be completely virtually regular if every submodule 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 $R$ with the unique maximal ideal $P$, we show that finitely presented (strongly) virtually regular modules are free if and only if $P$ is not principal; and that $P=Rp$ is principal if and only if finitely presented virtually regular modules are of the form $$R^n \oplus (\frac{R}{Rp})^{n_1} \oplus (\frac{R}{Rp^2})^{n_2} \oplus \cdots \oplus (\frac{R}{Rp^k})^{n_k}$$ for nonnegative integers $n,\,k,\,n_1,\,n_2,\cdots ,n_k.$ Similarly, we prove that $P=Rp$ is principal if and only if finitely presented strongly virtually regular modules are of the form $ R^n \oplus (\frac{R}{Rp})^{m}$, where $m,n$ are nonnegative integers. We also obtain that, $R$ admits a nonzero finitely presented completely virtually regular module $M$ if and only if $P=Rp$ is principal. Moreover, for a finitely presented $R$-module $M$, we prove that: $(i)$ if $R$ is not a DVR, then $M$ is completely virtually regular if and only if $M \cong (\frac{R}{Rp})^{m}$; and $(ii)$ if $R$ is a DVR, then $M$ is completely virtually regular if and only if $M\cong R^n \oplus (\frac{R}{Rp})^{m}.$ Finally, we obtain a characterization of finitely generated virtually regular modules over the ring of integers.