Alglat for modules over fsi rings and reflexivity.
thesisposted on 19.11.2015, 08:55 by Nicole Jane. Snashall
For a bimodule RMDelta where R and Delta are rings with unity, alglat RMDelta is the ring of all Delta-endomorphisms of M leaving invariant every R-submodule of M. The bimodule is said to be reflexive if the elements of alglat RMDelta are precisely the left scalar multiplications by elements of R. For most of the thesis Delta = R, a commutative ring with unity. However, in the early work, some results on the general structure of alglat are obtained, and in particular, Theorem 1.9 shows that it is an inverse limit. The next section of the thesis is concerned with reflexivity, and considers rings R for which all non-torsion or all finitely generated R-modules are reflexive. Theorem 3.4 gives eight equivalent conditions on an h-local domain R to the assertion that every finitely generated R-module is reflexive, that is R is scalar- reflexive. A local version of this property is introduced, and it is shown in Theorem 2.17 that a locally scalar-reflexive ring is scalar-reflexive. The remainder of this thesis considers alglat for all modules over an FSI ring. The local FSI rings are precisely the almost maximal valuation rings, and this is the first case to be settled. More details are then given of the structure of FSI rings and related rings. A completion is introduced in 6.4 to enable alglat to be determined for certain torsion modules over an indecomposable FSI ring. Theorem 7.3, in summarising the work of the last two chapters of the thesis, gives a complete characterisation of alglat for all modules over an FSI ring.