In my algebra textbook I came across the following:
Let $A$ be a $R$-algebra ($R$ is a commutative ring with unity $1_R$) and suppose $X\subseteq A$ a generator set for $A$.
In this context, what does a generator set for $A$ mean? Does it mean $A$ is generated by $X$ as a $R$-module?
Thanks.
It might be ambiguous, but I would usually parse it as a set of generators as an $R$-algebra, i.e. that every element of $A$ can be written as a polynomial in the elements of $X$ with coefficients in $R$.
For instance, $\{x\}$ generates $\Bbb Z[x]$ as a $\Bbb Z$-algebra, whereas you would need something like $\{1, x, x^2, \ldots \}$ to get a set of generators as a $\Bbb Z$-module.