Note that $R={\mathbb Z}$ is PID. So we have that if $I,\ J,\ K\subset R$ are ideals with $$ I = J\cap K$$ then $$(\ast)\ R/I = R/J\times R/K. $$
So we extend this. Let $R = {\mathbb Z}[x]$ which is UFD but not PID. Let $$ I= ((x^2+1)(x^3+1),2) = J\cap K,\ J =((x^2+1),2),\ K= ((x^3+1),2) $$
Then we have $(\ast)$ ? Right ?