On rings whose finitely generated faithful modules are generators

Abstract

A ring R over which every finitely generated faithful right R-module is a generator of the category mod-R of all right R-modules is called right FPF. FPF ring theory was initiated in this general setting by Faith in order to study from a unified point of view those rings that appear in Morita duality, commutative Prtifer rings and bounded Dedekind prime rings amongst others. The reader can consult Faith and Page’s book [ 141, where most of the basic results on FPF rings are contained. In Section 1 of this paper we construct semiprime FPF rings that are not semihereditary, thus answering [ 14, Question 111. It was an open question [l, p, 173171 whether or not the Pierce stalks of a semiprime FPF ring are FPF. We will answer this in the negative. We close Section 1 by proving that the maximal ring of quotients of a semiprime right FPF ring R satisfying a polynomial identity is the localization of R at the set of all nonzero divisors of the centre Z of R. In the case where R is module finite over Z this was obtained in [l, Proposition 1.91. Section 2 considers centres, Galois subrings, and group rings over FPF rings. By using the methods of Bergman and Cohn, cf. [4, Section 6.21, we show that every integrally closed commutative domain can be realized as the centre of a Bezout FPF domain, and conversely. This answers [14, Question 33 in the negative. We remark that a negative answer for nonsemiprime FPF rings is implicit in [24], where the authors construct a quasi-Frobenius ring whose centre is not quasi-Frobenius. We also provide additional examples showing that Galois subrings of semiprime FPF rings need not be FPF, cf. [14, Question 141. A positive result is that if C is a finite group of automorphisms of a reduced commutative FPF ring, then the fixed ring RG is FPF; this is fairly easy but our examples illustrate and limit the scope for possible generalizations. 425