Summary: The present paper is devoted to study of ring isomorphisms of \(\ast \)-subalgebras of Murray-von Neumann factors. Let \(\mathcal{M},\mathcal{N}\) be von Neumann factors of type II\(_1,\) and let \(S(\mathcal{M}),S(\mathcal{N})\) be the \(\ast \)-algebras of all measurable operators affiliated with \(\mathcal{M}\) and \(\mathcal{N}\), respectively. Suppose that \(\mathcal{A}\subset S(\mathcal{M})\), \(\mathcal{B}\subset S(\mathcal{N})\) are their \(\ast \)-subalgebras such that \(\mathcal{M}\subset\mathcal{A}\), \(\mathcal{N}\subset\mathcal{B} \). We prove that for every ring isomorphism \(\Phi:\mathcal{A}\to\mathcal{B}\) there exist a positive invertible element \(a\in\mathcal{B}\) with \(a^{-1}\in\mathcal{B}\) and a real \(\ast \)-isomorphism \(\Psi:\mathcal{M}\to\mathcal{N} \) (which extends to a real \(\ast \)-isomorphism from \(\mathcal{A}\) onto \(\mathcal{B}\)) such that \(\Phi(x)=a\Psi(x)a^{-1}\) for all \(x\in\mathcal{A} \). In particular, \( \Phi\) is real-linear and continuous in the measure topology. In particular, noncommutative Arens algebras and noncommutative \(L_{log} \)-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 \(S(\mathcal{M})\), which shows that the condition \(\mathcal{M}\subset\mathcal{A}\) is essential in the above mentioned result.