Ring Homomorphisms and Isomorphisms

In this paper all modules are left modules and all module homomorphisms act on the right. Ring homomorphisms are written on the left. If M is a module, let L(M) denote the of submodules of M. The Fundamental Theorem of Projective Geometry asserts that if D and AT are two division rings and X: L(D) -> L(K) is a isomorphism between two three-dimensional free modules, then X is induced by a semilinear isomorphism. This means that there is an additive isomorphism L: D -> KTM and a ring isomorphism a:D K such that (X)L = X(X) for each X e L(DTM) and (dV)L = a(d)(V)L for all V e DTM and deD. For convenience the phrase lattice isomorphism X: A -> 5 will be used to mean X: L(A) -> L(B) is a isomorphism. There has been some interest in generalizing this theorem to larger classes of rings. We prove here: