I've been reading a few texts and have been confused by some terminology regarding (sub)modules and (sub)rings. All definitions of submodules I've encountered have been in relation to the same ring so I am a little confused.
Let $R$ be a noetherian ring, $S$ a subring of $R$ and let $M$ be a finitely-generated module over $R$. What exactly does a (compact) $S$-submodule $N$ of $M$ mean?
For example, if $N$ were an $R$-submodule of $M$, then of course it would be finitely-generated since $R$ is noetherian. Am I right in thinking that if $N$ were an $S$-submodule of $M$ it would not necessarily be finitely-generated anymore? Would it be finitely-generated if we additionally assumed $N$ to be compact?
If $N_1, N_2$ were (compact) $S$-submodules of $R$-modules $M_1,M_2$ respectively, would $N_1 \oplus N_2$ be a (compact) $S$-submodule of $M_1 \oplus M_2$?
Same question as item 2. but for tensor products: $N_1 \otimes N_2$ as an $S$-submodule inside $M_1 \otimes M_2$.
Thanks in advance!
M
There is a concept called restriction of scalars and it works in the following way: suppose $f: S \to R$ is a ring homomorphism and $M$ is an $R$ module then $M$ can be equipped with a $S$ module structure by defining the scalar multiplication $s\cdot m:= f(s)m. $
Most probably, in your case $f$ is inclusion map and $N$ is apriori an $R$ submodule of $M.$