In module theory, let $A$ is a $C-$module and $Y$ is a non-empty subset of $A$. It is easy to prove that $B=\{ \sum_{y\in Y} cd | c\in C \}$ is the $C-$submodule of $A$ generated by $Y$.
Now, let $M$ is a $(R,S)-$bimodule and $X$ is any non-empty subset of $M$ and we define $N=\{ \sum_{x\in X} rxs | r\in R, s\in S \}$. Is $N$ a $(R,S)-$bisubmodule of $M$?
To prove $N$ is bisubmodule hence I need to show that:
- $N$ is left $R-$submodule of M
- $N$ is right $S-$submodule of M
- for any $r\in R, s\in S$ and $n\in N$ we have $r(ms)=(rm)s$ which is obvious since we have $n\in M$ for every $n\in N$
Now, this is what I've tried to solve number 1:
Let $n_1,n_2\in N$ and $r_1,r_2\in R$, we need to show $r_1n_1+r_2n_2\in N$. Since $n_1,n_2\in N$, we have $n_1=\sum_{x\in X} r'xs'$ for $r'\in R$ and $s'\in S$ and similarly $n_2=\sum_{x\in X} r''xs''$ for $r''\in R$ and $s''\in S$.
Then
$r_1n_1+r_2n_2=r_1\sum_{x\in X} r'xs'+r_2\sum_{x\in X} r''xs''=\sum_{x\in X} r_1r'xs'+\sum_{x\in X} r_2r''xs''$ $r_1n_1+r_2n_2=\sum_{x\in X}(r_1r'xs'+r_2r''xs'')$
Here I got stuck since I can't group them by $s'$ and $s''$ because the term in $r_1r'xs'$ and $r_2r''xs''$ would be different. I need to group them to produce $r_1n_1+r_2n_2=\sum_{x\in X} r'''xs'''$ so that $r_1n_1+r_2n_2$ would be in $N$.
(if it is done then number 2 would also be done since they are similar).
Or is there any counter-example that $N$ is not-necessary to be a $(R,S)-$bisubmodule? If it is true then I need the counter-example.
Here, I only want to derive the concept of "generated submodule by a non-empty subset" to "generated bisubmodule by a non-empty subset".
There is also the possibility that the set $N=\{ \sum_{x\in X} rxs | r\in R, s\in S \}$ in the context of bimodule theory shouldn't be used to substitute the set $B=\{ \sum_{y\in Y} cd | c\in C \}$ in the context of module theory. Since I try to defined $N$ myself hence it may highly incorrect.
Therefore I need the correct definition of "$(R,S)-$bisubmodule of $M$ generated by a non-empty subset $X$ of $M$" because I can't find reference on it.
You only need to allow finite multiplicity to get a bisubmodule by your definition.
It means you may have $r_1xs_1+r_2xs_2$ as an element of your bisubmodule without any force to represent it in singlton $r'xs'$ form.