Does free bimodule exist? Let $R,S$ be rings and $A$ be a set.
Does there exist a free $(R,S)$-bimodule $F(A)$ on $A$?
How do I construct it?
Is it just $\oplus_{i\in A} (R\times S)$?

An $(R,S)$-bimodule is the same thing as a left $R\otimes_{\Bbb{Z}}S^\mathrm{op}$ module, so the free $(R,S)$-bimodule on a set $A$ (exists and) is
$$\bigoplus_A R\otimes_{\Bbb{Z}}S^\mathrm{op}$$
