For a ring $R$, a faithful $R$-module $X$ is one where for all $r\in R$, either there exists $x\in X$ with $rx\neq 0$ or $r=0$. It is not in general true that every faithful $R$-module is free.
Let $R=\prod_{i=1}^n F_i$, where $F_i$ are fields. Is it then true that every faithful $R$-module is free?
At first I thought this would follow by inducing an action of each $F_i$ on $X$, but defining $f_ix:= (0,\ldots,f_i,\ldots,0)x$ doesn't necessarily result in an $F_i$-action because it's possible that $1\cdot x\neq x$. Ideas?