2020/01/28 by Tarizadeh, Abolfazl
#13A15 #13C10 #13C11 #14A05 #Commutative Algebra (math.AC) #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2001.10419
In this paper, new and significant advances on the understanding the structure of p.p. rings and their generalizations have been made. Especially 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, or equivalently, R/\mathfrakN is a p.p. ring and R_\mathfrakm is a primary ring for all \mathfrakm∈\rmMax(R). Some of the major results of the literature either are improved or are proven by new methods. In particular, we give a new and quite 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.