Characterizations of rings and modules by means of lattices.

Abstract

In this thesis we study the relationship between the lattice of submodules and the algebraic structure of a module. The key remark in our study will be the fact that the homomorphisms between two independent submadules of a module can be 'represented' by elements of its lattice of submoduleso Exploiting this fact we show that the endomorphism ring of a module which is the direct sum of more than three isomorphic submodules is determined up to isomorphism by its lattice of submodules. Lattice isomorphisms arise naturally in two ways, viz., through category equivalences and semi-linear isomorphisms. Any lattice isomorphism between a free module of infinite rank and a module containing at least one free submodule is shown to be induced by a category equivalence. This result is used to give new characterizations of Morita equivalence, If certain mild conditions are satisfied a lattice isomorphism between a free module of rank >3 and a faithful module is shown to give rise to a semi-linear isomorphism between the modules* If both nodules are free of rank n>3 then the question of whether there is a semi-linear isomorphism between them is equivalent to asking when an isomorphism. of matrix rings Rn Cý!! Sn implies a ring isomorphism R2ý S. -3- Wo study rings R with this property for any n and any ring S. The following are shown to be of this type (1) commutative rings (2) p-trivial rings (3) matrix rings over strongly regular rings left self-injective rings. Applying these results we give new examples of regular rings which uniquely co-ordinatize a complemented modular lattice of otder In particular we show such a co-ordinatization is always unique to within injective hull.