论文标题
冷凝域和$ d+xl [x] $ struction
Condensed domains and the $D+XL[X]$ construction
论文作者
论文摘要
令$ d $是商用字段$ k $的整体域，让$ \ natercal {i}％（d）$是$ d $的非零理想集。致电，对于$ i，j \ in \ mathcal {i}（d）$，如果$ ij = \ {ij = \ {ij | i \ in i，j \ in J \ in j \ in j \ in j \ in j \ call $ d $ condensed o a condensed域，则如果每对$ i，j $ i，j $ i，j $ ij $ ij $ condensed condensed condensed condensed。我们表明，如果$ a，b $是凝结域的元素，以至于$ ad \ cap bd = abd，$ then $（a，b）=D。$。代数15（1987），1895- 1920年]，一个固定域是一个$ \ ast $ domain，即$ d $满足$ \ ast：$ \ ast：每对$ $ \ {a_ {a {i} _ {i = 1}^{m}，\ {b_ {j} _ { （a_ {i} b_ {j}）。$我们表明，当$ d $是$ \ ast $ - domain时，当凝结域$ d $是预先切换的。我们还表明，如果$ a \ subseteq b $是域的扩展和$ a+xb [x] $的浓缩，那么$ b $必须是一个字段，必须凝结$ a $，在这种情况下，在这种情况下，在这种情况下，$ [b：k] <4。指出，如果$ d $不是字段$ d [x] $永远不会被凝结。因此，对于$ d $凝结的$ d+xk [x] $是一种从旧构建新的冷凝域的方式
Let $D$ be an integral domain with quotient field $K$ and let $\mathcal{I}% (D) $ be the set of nonzero ideals of $D$. Call, for $I,J\in \mathcal{I}(D)$ , the product $IJ$ of ideals condensed if $IJ=\{ij|i\in I,j\in J\}.$ Call $D$ a condensed domain if for each pair $I,J$ the product $IJ$ is condensed. We show that if $a,b$ are elements of a condensed domain such that $aD\cap bD=abD,$ then $(a,b)=D.$ It was shown in [Comm. Algebra 15 (1987), 1895-1920] that a pre-Schreier domain is a $\ast $-domain, i.e., $D$ satisfies $\ast :$ For every pair $\{a_{i}\}_{i=1}^{m},\{b_{j}\}_{j=1}^{n}$ of sets of nonzero elements of $D$ we have $(\cap (a_{i}))(\cap b_{j})=\cap (a_{i}b_{j}).$ We show that a condensed domain $D$ is pre-Schreier if and only if $D$ is a $ \ast $-domain. We also show that if $A\subseteq B$ is an extension of domains and $A+XB[X]$ is condensed, then $B$ must be a field and $A$ must be condensed and in this case $[B:K]<4.$ In particular we study the necessary and sufficient conditions for $D+XL[X]$ to be condensed, where $D$ is a domain and $L$ an extension field of $K.$ It may be noted that if $D$ is not a field $D[X]$ is never condensed. So for $D$ condensed $D+XK[X]$ is a way of constructing new condensed domains from old
