|dc.description.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
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
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.
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.||en_US
|dc.title||Characterizations of rings and modules by means of lattices.||en_US
|dc.rights.holder||The copyright of this thesis rests with the author and no quotation from it or information derived from it may be published without the prior written consent of the author||