A more structural proof using homomorphisms and similar tools that every ideal of $M_n(R)$ is of the form $M_n(I)$ Let $R$ be a ring with unity , we know that if $J$ is an ideal of $M_n(R)$ then for some ideal $I$ of $R$ , $J=M_n(I)$ . The proof I know is very tedious and uses laborious manipulations using elementary matrices and all that . I was thinking , can we give a more structural proof , like using ring homomorphisms or so ? I kind of feel some hint of correspondence theorem in the result but cannot actually grasp it ... Please help

> I kind of feel some hint of correspondence theorem in the result but cannot actually grasp it

Well, there is a correspondence theorem going on, but I'm not sure that it will simplify matters.

The relationship is established through Morita's theorems, and this is just a special case. If you dig down into it, it will tell you the answer in terms of homomorphisms.

The "very tedious... laborious" version is more accessible, by comparison, and probably can be streamlined a bit to be tolerable.