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, $R/\mathfrak{ N}$ is a p.p. ring and $R_{ \mathfrak{ m}}$ is a primary ring for all $\mathfrak{ m}\in\rm{ Max}(R)$. 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.
Categories: math.AC math.RA