In The Many Lives of Lattice Theory Gian-Carlo Rota says the following.
Necessary and sufficient conditions on a commutative ring are known that insure the validity of the Chinese remainder theorem. There is, however, one necessary and sufficient condition that places the theorem in proper perspective. It states that the Chinese remainder theorem holds in a commutative ring if and only if the lattice of ideals of the ring is distributive.
The essay can be found here, and the quotation comes from the third page in the file.
I know the following version of the Chinese remainder theorem for rings (not necessarily commutative).
Suppose $R$ is a ring and $A,A_1, \ldots,A_k$ are ideals of $R.$ If
$(1)$ $A_1 \cap \ldots \cap A_k = A,$ and
$(2)$ $A_i + A_j = R$ for all $1 \leq i < j \leq k,$
then $R/A \cong R/A_1\times\ldots\times R/A_k$ via an isomorphism which is both a ring isomorphism and an $R$-module isomorphism.
This version of the theorem comes from these lecture notes.
Clearly, there are some lattice-theoretic conditions on the ideals here, but I don't understand what G.C. Rota means by "the Chinese remainder theorem". He cannot mean this version because it holds for any rings. Could you give me the exact wording of the theorem he mentions? Also, can I find its proof anywhere? And if it's possible, could you explain to me why (or if) commutativity is important in the theorem he mentions?