  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.
