Projective modules over the endomorphism ring
of a biuniform module
by Gena Puninski
The well-known Kaplansky's theorem states that every projective module over
a local ring is free. By Facchini, the endomorphism ring of a biuniform module
has at most two right (left and two-sided ideals). Here a module M is biuniform
if every two nonzero submodules of M has a nonzero intersection, and the
sum of two proper submodules of M is proper.
Is it possible to generalize Kaplansky's theorem? Dung and Facchini showed
that every finitely generated projective module over the endomorphism ring
of a biuniform module is free. Recently Puninski found an example of indecomposable
projective non-free module over the endomorphism ring of a uniserial module.
In this talk we give a presice criterion for when every projective
right module over the endomorphism ring of a biuniform module is free.
We also discuss the general problem of classification of projective modules
over this class of rings. We give examples of a uniserial module M, such
that the endomorphism ring of M is a distributive ring do not admitting
classical localization. This answers a question by Tuganbaev.