Chapter 3 - Nonsingular FPF rings

Summary

In this chapter we study nonsingular FPF rings. For commutative rings we have: If R is FPF, then Q_c = Q_cl(R) is FPF, Faith [78, Th. C]. For general noncommutative rings the question remains open. In case R is right nonsingular or semiprime we show that Q_m = Q^r_max(R) is FPF (both sides) if R is FPF (one side). In fact, under these circumstances we give a complete description of Q_m(R) as a regular ring of bounded index. Using this description we are able to show prime right FPF rings are Goldie on both sides and that every right Goldie right FPF ring is left Goldie, too. We also show that right FPF rings are right semihereditary iff left semihereditary iff they and all their matrix rings are Baer rings. Finally we include some results on fully idempotent rings and in particular V-rings which are FPF.

Below and hereafter, ^⊥X(resp. X^⊥) denotes the left annihilator left (resp. right) ideal corresponding to a nonempty subset X of R.

PROPOSITION.

Let R be right FPF. Then:

If M is a finitely generated and faithful right R-module, then the dual module

If I and K are right ideals such that I ∩ K = 0, then ^⊥IR + ^⊥KR = R, and hence ⊥I + ^⊥K generates R-mod. Thus, either^⊥I ≠ 0, or^⊥K ≠ O.