Simple rings with a one-sided minimal ideal may be represented as Rees matrix rings, and conversely. The latter are defined as $I \times \Lambda$ - matrices over a division ring with only a finite number of nonzero entries with certain addition and multiplication. For Rees matrix rings we construct here their isomorphisms, their translational hulls and isomorphisms of the translational hulls, all this in terms of certain type of matrices of arbitrary size over division rings. We also study $r$-maximal Rees matrix rings. This theory runs parallel to that of Rees matrix semigroups.