$R[\alpha,\beta] = R[\alpha][\beta]$ true? Is it true that for a ring $R$,  $R[\alpha,\beta] = (R[\alpha])[\beta]$? I know this is true for field extensions, but I'm not completely sure it's true for ring extensions.
 A: $\newcommand{\monic}{\rightarrowtail}$
Once $R[\alpha]$, $R[\alpha,\beta]$ and $R[\alpha][\beta]$ are defined it's true by abstract nonsense.

Let $\alpha$ and $\beta$ be drawn from $S$, i.e., we have the implicit extension $R\monic S$. Define a "category of $S/R$ intermediates" with 


*

*objects the diagrams $R\stackrel{i_A}{\monic} A \stackrel{j_A}{\monic} S$ (or "just $A$") and

*morphisms those $f:A\monic B$ such that $i_B=fi_A$ and $j_A=j_Bf$.


Next, define


*

*$R[\alpha]$ to be initial among $S/R$ intermediates $A$ such that $\alpha \in A$,

*$R[\alpha][\beta]$, initial among $S/R[\alpha]$ intermediates $A$ such that $\beta \in A$, and

*$R[\alpha,\beta]$, initial among $S/R$ intermediates $A$ such that $\alpha\in A$ and $\beta \in A$.


Then


*

*since $\alpha\in R[\alpha,\beta]$, there is a unique morphism of $S/R$ intermediates $R[\alpha]\rightarrowtail R[\alpha,\beta]$ by the definition of $R[\alpha]$, so $R[\alpha,\beta]$ lifts to an $S/R[\alpha]$ intermediate $R[\alpha]\monic R[\alpha,\beta] \monic S$;

*since $\beta\in R[\alpha,\beta]$, there is a unique morphism of $S/R[\alpha]$ intermediates $\phi: R[\alpha][\beta] \monic R[\alpha,\beta]$ by the definition of $R[\alpha][\beta]$; 

*since $R[\alpha][\beta]$ descends to the $S/R$ intermediate $R\monic R[\alpha][\beta]\monic S$ by composition with $R\monic R[\alpha]$, and since $\alpha,\beta\in R[\alpha][\beta]$, there exists a unique morphism of $S/R$ intermediates $\psi: R[\alpha,\beta]\monic R[\alpha][\beta]$.


Finally,


*

*by the definition of $R[\alpha]$, $\psi$ lifts to a morphism of $S/R[\alpha]$ intermediates, so we can compose to get a morphism $\psi\phi:R[\alpha][\beta]\monic R[\alpha][\beta]$, which must be the identity by the definition of $R[\alpha][\beta]$;

*by composition with $R\monic R[\alpha]$, $\phi$ descends to a morphism of $S/R$ intermediates, so we can compose toget a morphism $\phi\psi: R[\alpha,\beta]\monic R[\alpha,\beta]$, which must be the identity by the definition of $R[\alpha,\beta]$.


Altogether we have shown that the $\phi$ and $\psi$ defined above constitute a canonical isomorphism
$$R[\alpha,\beta]\simeq R[\alpha][\beta]\text{.}$$
