Let $R\hookrightarrow S$ be Noetherian (noncommutative) rings without zero divisors with $\mathrm{rk}\_{R} S < \infty$ (e.g. $S=R\*G$ the crossed product of $R$ with a finite group $G$). Let $M$ be a finitely generated $S$-module.


What are sufficient criteria we can impose on $R \hookrightarrow S$ that $\mathrm{rk}\_R M$ is well behaved, e.g. $$\mathrm{rk}\_R M = \mathrm{rk}\_S M \cdot \mathrm{rk}\_R S$$ or alternatively $$\mathrm{tor}\_R M = \mathrm{tor}\_S M?$$ What are examples where this fails?



$\mathrm{rk}_R M$ vs $\mathrm{rk}_S M$ - how nice need $R,S$ be?
Two for my applications important cases turned out to have quite easy solutions. For the sake of completeness I'll post them, although I'm still wondering if there is a more general theory for ranks under base change.


In one case $x \mapsto x^{p^k}$ provided a map $S \rightarrow R$, hence $\mathrm{tor}\_R M = \mathrm{tor}\_S M$.


In one other case it held that $M$ had finite projective dimension over $S = R\*G$ and that $S$ was a local ring, hence $M$ had a free resolution of finite length $$0 \rightarrow S^{d\_n} \rightarrow \dots \rightarrow S^{d\_0} \rightarrow M \rightarrow 0.$$ As $S$ is a free $R$-module, this is also a free resolution of $M$ as an $R$-module and the rest follows by counting ranks.


