论文标题
非唯一分解域II的独特分解特性II
Unique factorization property of non-unique factorization domains II
论文作者
论文摘要
令$ d $为一个整体域。如果有$ v $ $ d $的估值，则非零的nonunit $ a $ a $ a $ d $，称为估值元素。我们说，如果每个非零零单元的$ d $可以写为估值元素的有限产品，则$ d $是一个估值分解域（VFD）。在本文中，我们研究了VFD的一些环理论特性。除其他外，我们表明（i）vfd $ d $是Schreier，因此，$ {\ rm cl} _t（d）= \ {0 \ {0 \} $，（ii）如果$ d $是p $ v $ md，则$ v $ md，则$ d $是vfd，如果$ d $是fly d $，则只有$ d $ d $ d $ d $ d $ d $ d $ dirial dirial，如果是dirial dirial dirial divial nf and if dif。 $ d $，是VFD，（iii）vfd $ d $是一个弱阶段的GCD域，并且仅当$ d $是Archimedean。我们还研究VFD的独特分解属性。
Let $D$ be an integral domain. A nonzero nonunit $a$ of $D$ is called a valuation element if there is a valuation overring $V$ of $D$ such that $aV\cap D=aD$. We say that $D$ is a valuation factorization domain (VFD) if each nonzero nonunit of $D$ can be written as a finite product of valuation elements. In this paper, we study some ring-theoretic properties of VFDs. Among other things, we show that (i) a VFD $D$ is Schreier, and hence ${\rm Cl}_t(D)=\{0\}$, (ii) if $D$ is a P$v$MD, then $D$ is a VFD if and only if $D$ is a weakly Matlis GCD-domain, if and only if $D[X]$, the polynomial ring over $D$, is a VFD and (iii) a VFD $D$ is a weakly factorial GCD-domain if and only if $D$ is archimedean. We also study a unique factorization property of VFDs.
