Given a regular local ring $(R,m)$ and a finitely generated free $R$-algebra $S$. Let $M$ be a left $S$-module of finite length, $\ell_S(M)=r<\infty$.

Under what conditions is $\ell_R(M)<\infty$? If this is the case, can we compute $\ell_R(M)$ in terms of $\ell_S(M)$?

For example if $S=M_n(R)$, then i think we have $\ell_R(M)=n\ell_S(M)$.

If $S$ is commutative and local, with maximal ideal n, then according to Liu one has the following: $\ell_R(M)=[S/n:R/m]l_S(M)$.

Are there general formulas for length and "restriction of scalars"? I'm especially interested in the case when $S$ is a maximal $R$-order in a division algebra. Literature tips are also appreciated.

Given a regular local ring $(R,m)$ and a finitely generated $R$-algebra $S$, which is free as an $R$-module. Let $M$ be a left $S$-module of finite length, $\ell_S(M)=r<\infty$.

Under what conditions is $\ell_R(M)<\infty$? If this is the case, can we compute $\ell_R(M)$ in terms of $\ell_S(M)$?

For example if $S=M_n(R)$, then i think we have $\ell_R(M)=n\ell_S(M)$.

If $S$ is commutative and local, with maximal ideal n, then according to Liu one has the following: $\ell_R(M)=[S/n:R/m]l_S(M)$.

Are there general formulas for length and "restriction of scalars"? I'm especially interested in the case when $S$ is a maximal $R$-order in a division algebra. Literature tips are also appreciated.