  The present paper is devoted to study of ring isomorphisms of
$\ast$-subalgebras of Murray--von Neumann factors. Let $\cM,$ $\cN$ be von
Neumann factors of type II$_1,$ and let $S(\cM),$ $S(\cN)$ be the
$\ast$-algebras of all measurable operators affiliated with $\cM$ and $ \cN,$
respectively. Suppose that $\cA\subset S(\cM),$ $\cB\subset S(\cN)$ are their
$\ast$-subalgebras such that $\cM\subset \cA,$ $\cN\subset \cB.$ We prove that
for every ring isomorphism $\Phi: \cA \to \cB$ there exist a positive
invertible element $a \in \cB$ with $a^{-1}\in \cB$ and a real
$\ast$-isomorphism $\Psi: \cM \to \cN$ (which extends to a real
$\ast$-isomorphism from $\cA$ onto $\cB$) such that $\Phi(x) = a\Psi(x)a^{-1}$
for all $x \in \cA.$ In particular, $\Phi$ is real-linear and continuous in the
measure topology. In particular, noncommutative Arens algebras and
noncommutative $\cL_{log}$-algebras associated with von Neumann factors of type
II$_1$ satisfy the above conditions and the main Theorem implies the automatic
continuity of their ring isomorphisms in the corresponding metrics. We also
present an example of a $\ast$-subalgebra in $S(\cM),$ which shows that the
condition $\cM\subset \cA$ is essential in the above mentioned result.
