几乎v-整环的局部化与拉回图研究

Characterization of the Localization of an Almost v-Domain and Its Pullback

引入了几乎v-整环的概念.举例说明了几乎v-整环的局部化不一定是几乎v-整环.证明了若{Rα}是整环R的平坦扩环且R=∩Rα具有局部有限特征,如果Rα都是几乎v-整环,则R也是几乎v-整环.也研究了关于几乎v-整环的Nagata型定理.最后研究了几乎v-整环在(ΔM)型拉回图中的性质,证明了在(ΔM)型拉回图中,整环R是几乎v-整环当且仅当整环D和T都是几乎v-整环且TM是AV-整环.特别地,给出了若k是整环D的商域,则D+Xk[X](或D+Xk[[X]])是几乎v-整环当且仅当D是几乎v-整环.

In this paper we introduce the notion of almost v-domain. We show that a quotient ring of an almost v-domain is not necessarily an almost v-domain. We indicate that let { Rα} be a family of flat overrings of R with R = ∩ Rα locally finite, if each of Rα is an almost v-domain, then R is an almost v-domain. We also show that in the pullback diagram of type （ △M ） , R is an almost v-domain if and only if D and T are almost v-domains and TM is an AV-domain. As a corollary, we prove that let k be the quotient field of an integral domain D, then D ＋ Xk[ X] （ resp. D ＋ Xk[ [ X]] ） is an almost v-domain if and only if D is an almost v-domain.