Lattice definability of certain matrix rings

Let $R=M_n(K)$ be the ring of square matrices of order $n\geqslant 2$ over the ring $K= \mathbb{Z}/p^k\mathbb{Z}$, where $p$ is a prime number, $k\in\mathbb{N}$. Let $R'$ be an arbitrary associative ring. It is proved that the subring lattices of the rings $R$ and $R'$ are isomorphic if and only if the rings $R$ and $R'$ are themselves isomorphic. In other words, the lattice definability of the matrix ring $M_n(K)$ in the class of all associative rings is proved. The lattice definability of a ring decomposable into a direct (ring) sum of matrix rings is also proved. The results obtained are important for the study of lattice isomorphisms of finite rings.
Bibliography: 13 titles.