Consider the ring $$R = \frac{\mathbb{C}[X,Y]}{ (Y^3 - X^3(X+1))} = \mathbb{C}[x,y]. $$ Let $t = y/x$. How can I show that $t \notin R$?
Also is, it true that $R[t] = \mathbb{C}[t]$?
I have $y^3 = x^3 (x+1)$, hence $t^3 = x+1$ and so $x = t^3 - 1$. Then $$y = tx = t(t^3 - 1). $$ So I would say that $$R[t] = \mathbb{C}[x,y,t] = \mathbb{C}[t^3 - 1, t(t^3-1), t] = \mathbb{C}[t]. $$ Is this reasoning correct?
For your first question, assume for a contradiction that there exists an element $t \in R$ such that $$xt = y.$$ This means that there exist $T, F \in \mathbf{C}[X,Y]$ such that $$XT = Y + (Y^3-X^3(X+1))F.$$
Can that happen? (Hint: look at it mod $X$)
Your reasoning for $R[t] = \mathbf C[t]$ looks good.