I can't seem to find a solid definition of the subalgebra generated by a set anywhere.
Let $A$ be a commutative ring, $M$ an $A$-algebra and $S$ an $A$-submodule of $M$. We call $S$ a $subalgebra$ of $M$ if $S$ is also a subring of $M$.
The $A$-submodule $(S)$ of $M$ generated by a subset $S\subseteq M$ is the smallest $A$-submodule of $M$ which contains $S$. It is equal to the set of all finite $A$-linear combinations of elements in $S$.
The subring $(S)$ of $M$ generated by the subset $S\subseteq M$ is the smallest subring of $M$ which contains $S$. It is equal to the set of all finite sums of finite products of elements of $S$.
I'm guessing, then, that the subalgebra $(S)$ generated by $S\subseteq M$ is going to be the smallest subalgebra of $M$ which contains $S$? If so, am I right in saying that $(S)$ is equal to the set of all finite $A$-linear combinations of finite products of $S$?
Thanks!
Your definition is correct. Another way to think of the subalgebra of the $A$-algebra $M$ generated by $S\subset M$ is polynomials in elements of $S$ with coefficients in $A$.