Which ring is $R[X,Y,Z,T]/(X-Y^2,T-Y^4,T^3-Z)$ isomorphic to? 
Which ring is $R[X,Y,Z,T]/(X-Y^2,T-Y^4,T^3-Z)$ isomorphic to?

I already did substitution for $X$ so we get the ring $R[x,x^{1/2},x^6,x^2]$ but I don't know to which ring this is isomorphic. 

Note that $T^3 - Y^{12}$ divisible by $T- Y^4$, so we have the equality of ideals
$$\langle T- Y^4, T^3 - Z\rangle = \langle T- Y^4,  T^3 - Y^{12}, T^3 - Z\rangle =\langle T- Y^4, Z- Y^{12}\rangle$$
and so
$$\langle X-Y^2, T- Y^4, T^3 - Z\rangle = \langle X-Y^2, T- Y^4, Z- Y^{12}\rangle$$
Now the evaluation map
$R[Y,X,Z,T]\to R[Y]$,
$X\mapsto Y^2$, $Z\mapsto Y^{12}$, $T\mapsto Y^4$ gives the isomorphism
$$R[Y,X,Z,T]/\langle X-Y^2, T- Y^4, Z- Y^{12}\rangle \simeq R[Y]$$\
