Structure theory of p.p. rings and their generalizations

Abstract

In this paper, new and significant advances on the understanding the structure of p.p. rings and their generalizations have been made. Specially among them, it is proved that a commutative ring R is a generalized p.p. ring if and only if R is a generalized p.f. ring and its minimal spectrum is Zariski compact. We also prove that the total ring of fractions of a given ring R is absolutely flat if and only if R is reduced, its minimal spectrum is Zariski compact and every finitely generated and faithful ideal of R admits a non zero-divisor. Some of the major results of the literature either are improved or are proven by new methods. In particular, we give a new and elementary proof to the fact that a commutative ring R is a p.p. ring if and only if R[x] is a p.p. ring. We also prove the same assertion for p.f. rings. Finally, the new notion of fluffy ring is introduced and studied which generalizes at once both generalized p.p. ring and almost p.p. ring notions.