Suppose I have a ring $R$ whose underlying group is $G$.
Now suppose I read the definition:
Let $M$ be an $n$-dimensional left module over $R$.
Is this a conventional way of defining $M$ as being $G^n$ over $R$ where $G^n$ is a repeated direct product of $G$ with itself?
Also: a module's abelian group need not be the same group underlying its ring, correct?