Semiperfect CFPF rings.

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.