Semiperfect CFPF rings.
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Date
1987
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Abstract
The Wedderburn-Artin Theorem (1927) characterised
semisimple Artinian rings as finite direct products of
matrix rings over division rings. In attempting to
generalise Wedderburn's theorem, the natural starting point
will be to assume R/RadR is semisimple Artinian. Such
rings are called semilocal. They have not been completely
characterised to date. If additional conditions are
imposed on the radical then more is known about the
structure of R. Semiprimary and perfect rings are those
rings in which the radical is nilpotent and T-nilpotent
respectively. In both these cases the radical is nil, and
in rings in which the radical is nil, idempotents lift
modulo the radical. Rings which have the latter property
are called semiperfect. The characterisation problem of
such rings has received much attention in the last few
decades.
We study semiperfect rings with a somewhat strong condition
arising out of the status of generators in the module
categories. More specifically, a ring R is CFPF iff every
homomorphic image of R has the property that every finitely
generated faithful module over it generates the
corresponding module category.
The objective of this thesis is to develop the theory that
leads to the complete characterisation of semiperfect right
CFPF rings. It will be shown (Theorem 6.3.17) that these
rings are precisely finite products of full matrix rings
over right duo right VR right a-cyclic right CFPF rings.
As far as possible theorems proved in Lambek [16] or Fuller
and Anderson [12] have not been reproved in this thesis and
these texts will serve as basic reference texts.
The basis for this thesis was inspired by results contained
in the first two chapters of the excellent LMS publication
"FPF Ring Theory" by Carl Faith and Stanley Page [11]. Its
results can be traced to the works of G. Azumaya [23], K.
Morita [18], Nakayama [20;21], H. Bass [4;5],
Carl Faith [8;9;10], S. Page [24;25] and B. Osofsky [22].
Our task is to bring the researcher to the frontiers of FPF
ring theory, not so much to present anything new.
Description
Thesis (M.Sc.)-University of Durban-Westville, 1987.
Keywords
Molecules (Algebra), Rings (Algebra), Theses--Mathematics.