Q:

The ring $K[t^2,t^3]$ is not a PID

I have to show that the ring $R=K[t^2,t^3]$ is not a PID, where $K$ is a field. 

Consider the ideal $I=(t^2,t^3)$. If $R$ is a PID then there exist $f(t^2,t^3)\in R$ such that $(t^2,t^3)= (f(t^2,t^3))$. Since, $t^2\in I\implies t^2=f(t^2,t^3)g(t^2,t^3) \implies $ either $f$ is constant polynomial or $f$ is of degree 2 polynomial.  If it is degree 2 polynomial then it can't give $t^3$ but what if it is constant?  

A:

Here is another take. 
Every PID is a UFD. 
$a=t^2$ and $b=t^3$ are irreducible elements of $R$ but $a^3=b^2$ contradicts unique factorization. 
Therefore, $R$ is not a UFD and so cannot be a PID.