The converse to CRT asks: does $R/(I \cap J)\simeq R/I \times R/J$ imply $I+J=R$? For me $\simeq$ is an abstract ring isomorphism, sending $1$ to $1$, not necessarily $R$ linear, i.e., one of $R$--Modules.
The choice $R=\prod {\mathbb Z}$, $I=J=0$, from previous answers show that the answer is no in the case of commutative rings with 1. But what about domains? If this has been answered before, sorry, but I do not think so.
Thanks for the correction of the misprint.