A ring R of linear transformations of a vector space M over a division ring D is called distinguished iff (1) the lattice J of all R-submodules of M is a distributive sublattice of the lattice L of all subspaces of M, and (2) the set of all linear transformations of M leaving J invariant is R. The study of such rings is motivated by a paper of Wolfson [2] in which J is a chain and by several recent papers of Behrens [4]. Behrens studies Artinian rings R with unity having faithful Rmodules M such that the lattice of R-submodules of M is distributive. Our primary interest is with distinguished rings R for which J is finite as well as distributive. Such a condition does not force the ring R to be Artinian. A basic tool in our study is a lattice theorem (1.1) stating that every element of J is a direct sum of elements of L associated with the irreducible elements of J. It is shown that every finite distributive sublattice of L containing 0 and M is the lattice of submodules of a distinguished ring R. If D has characteristic 0, then a finite sublattice of L must be distributive in order to be the lattice of submodules of a ring of linear transformations of M. A subspace N of M is called J-distributive iff N rl (A u B) = (N rl A) u (N rl B) for all A, B E J. It is shown that N is J-distributive if N = Me for some idempotent e E R. All subspaces of M are J-distributive iff J is a chain. Wolfson proved that R is a Baer ring (i.e., every annihilating right or left ideal of R is generated by an idempotent) if J is a chain. We show that this is almost the only case in which a distinguished ring is a Baer ring. Every distinguished ring R is a direct sum of subrings of the form eiRej, when 1 = e, + . + e,, (direct sum), each eiRej is a full ring of linear transformations, eiRe =O if i < j, and 1i * j eiRej is the radical of R. Two distinguished rings are shown to be isomorphic iff their vector spaces are related in an obvious way.