Since it's been a while since I did my course in basic abstract algebra, as I am playing around with homemade examples trying to revise stuff and making sure that I still get the very most fundamental, I just wanted to know if the following is legitimate.
Consider the two rings $R_1 = \mathbb{C}[x,y]/(x+2y^2)$ and $R_2 = \mathbb{C}[s,t,u]/(st + u^2)$. I make the claim that $R_1$ is isomorphic to a subring of $R_2$ by means of the injective morphism $f : R_1 \rightarrow R_2$ defined by $$x \mapsto st , \qquad y \mapsto \frac{1}{\sqrt{2}} u .$$ Then, it is clear that $f(x)+2(f(y))^2 = 0$.
Is this a legitimate proof and course of action, or is there some silly mistake that I am making?
Thank you in advance.
Yes, it is legitimate, but I would say it this way: It is a property of polynomial algebras that this grants you a homomorphism $f:\mathbb C[x,y]\to R_2$ given the choices you made.
The first homomorphism theorem says that $\mathbb C[x,y]/\ker(f)\cong \mathrm{Im}(f)\subseteq R_2$.
The only thing left for you to decide is whether or not $\ker(f)=(x+2y^2)$. Certainly by your choice, you have already guaranteed that $(x+2y^2)\subseteq \ker(f)$, but proving the reverse containment would be necessary for complete success.