论文标题
P.P.的结构理论戒指及其概括
Structure theory of p.p. rings and their generalizations
论文作者
论文摘要
在本文中，了解P.P.的结构的新的和重大的进步戒指及其概括。特别是其中，事实证明，可交换的环$ r $是广义的P.P.且仅当$ r $是广义的P.F.戒指及其最小频谱是Zariski Compact，或等效地，$ R/\ Mathfrak {N} $是P.P. ring and $ r _ {\ mathfrak {m}} $是所有$ \ mathfrak {m} \ in \ rm {max}（r）$的主要环。文献的一些主要结果要么通过新方法得到改进或得到证明。特别是，我们给出了一个新的，相当基本的证据，证明了一个通勤的环$ r $是P.P.且仅当$ r [x] $是P.P.戒指。
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.
