A look at FPF rings

Abstract

An error in Corollary 9.32 of [C. Faith, Rings and Things and a Fine Array of Twentieth Century Associative Algebra, Mathematical Surveys and Monographs, Vol. 65 (American Mathematical Society, Providence, RI, 2004)], motivated us to consider again FPF rings which were initiated by Faith in the 1970s. In this paper, it is shown that a commutative ring [Formula: see text] is reduced FPF if and only if it is [Formula: see text]-semihereditary. We show that when a semiperfect ring with a strongly right bounded basic ring with right and left Ore conditions, is an FPF ring. After some general results, the article focuses on rings of continuous functions. We give some algebraic characterizations for a [Formula: see text] to be FPF and retrieve a result of Jorge Martinez. Also, we show that a space [Formula: see text] is fraction-dense if and only if [Formula: see text] is a continuous ring.