Update: I posted the question with the WRONG inequality!! (Now it's ok)
I have a problem with the following statement:
True or False: Let $S$ be a subring of $R$ (unitary but not necessarily commutative), $M$ a $\textbf{free}$ $R$-module and suppose that $\mathrm{rank}_R M, \mathrm{rank}_S M$ both exist, then $\mathrm{rank}_R M\leq \mathrm{rank}_S M$.
I think it's true (but maybe is false). My thoughts: $R$ is a $S$-module with $s\cdot r=sr$, so if $\mathrm{rank}_R M=r$ and $\mathrm{rank}_S M=s$ then $M\cong \bigoplus_r R$ (isomorphism of $R$-modules) and $M\cong \bigoplus_s S$ (isomorphism of $S$-modules). Moreover, we know that the inclusion $\iota:R\to S$ induces an injective $S$-module homomorphism $\bigoplus_r S\to \bigoplus_r R$ and $\bigoplus_s S\to \bigoplus_s R$ but I cannot relate these things.
Can anyone provide me some hints? Thank you!
Let $S$ be a field and let $R = S \oplus S$, $M = S \oplus S \oplus S \oplus S$. Then $S \leq R$ and $\operatorname{rank}_S(M) = 4$, but $\operatorname{rank}_R(M) = 2$.