The direct sum of two finitely generated algebras is finitely generated Let $X$ and $Y$ be sets of variables. Let $R$ be a commutative ring. Suppose $S$ and $T$ are finitely generated $R$-algebras. Then $S\cong R[X]/I$ and $T\cong R[Y]/J$ for some $R[X]$-ideal $I$ and $R[Y]$-ideal $J$. 
Why is $S\bigoplus T$ a finitely generated $R$-algebra? I wanted to say "just combine the generators of $S$ and $T$" together, but isn't that saying that
$$\frac{R[X]}{I}\oplus\frac{R[Y]}{J}\cong \frac{R[X,Y]}{IR[X,Y]+JR[X,Y]}?$$
But if that's the case, then I'm confused because I thought
$$\frac{R[X]}{I}\otimes\frac{R[Y]}{J}\cong \frac{R[X,Y]}{IR[X,Y]+JR[X,Y]}?$$
But I thought the direct sum and tensor product were very different operations?

What do you mean by $\oplus$ when you are working with $R$-algebras, or more generally with commutative rings? See this wiki section.
If you are writing $\oplus$ to mean the coproduct (in the category of $R$-algebras), then you ought to write $\otimes_R$ instead, and your equation is correct. If you mean the product, then you ought to write 
$\times$ instead, and your equation is incorrect (e.g., with $I=J=0$, $\mathbb{C}[X]\times \mathbb{C}[Y]$ is not a domain, so it can't be $\mathbb{C}[X,Y]$), but see @Lubin's answer for more in this case. If you mean neither of those, you should clarify what it is that you mean (and your equation is probably incorrect).
