论文标题
主要避免物业
Prime Avoidance Property
论文作者
论文摘要
让$ r $为可交换的戒指，我们说$ \ nathcal {a} \ subseteq spec（r）$具有主要的回避属性，如果$ i \ subseteq \ bigcup_ {p \ in \ nathcal {a} a}} p $，则适用于理想的$ r $，然后$ i $ a $ a $ a \ p \ r $ P $。我们准确地确定$ \ Mathcal {a} \ subseteq spec（r）$具有主要回避属性。特别是，如果$ \ Mathcal {a} $具有主要的回避属性，则$ \ Mathcal {a} $是紧凑的。对于某些经典戒指，我们显示了匡威（例如Bezout Rings，$ QR $ - 域，零维环和$ C（x）$）。我们举例说明紧凑型集合$ \ mathcal {a} \ subseteq spec（r）$，其中$ r $是一个prufer域，没有$ p.a $ -property。最后，我们表明，如果$ v，v_1，\ ldots，v_n $是字段$ k $和$ k $和$ v [x] \ nsubseteq \ bigcup_ {i = 1}^n v_i $ for K $ in K $中的某些$ x \，则在v $ v+v+x \ bigIn中存在。
Let $R$ be a commutative ring, we say that $\mathcal{A}\subseteq Spec(R)$ has prime avoidance property, if $I\subseteq \bigcup_{P\in\mathcal{A}}P$ for an ideal $I$ of $R$, then there exists $P\in\mathcal{A}$ such that $I\subseteq P$. We exactly determine when $\mathcal{A}\subseteq Spec(R)$ has prime avoidance property. In particular, if $\mathcal{A}$ has prime avoidance property, then $\mathcal{A}$ is compact. For certain classical rings we show the converse holds (such as Bezout rings, $QR$-domains, zero-dimensional rings and $C(X)$). We give an example of a compact set $\mathcal{A}\subseteq Spec(R)$, where $R$ is a Prufer domain, which has not $P.A$-property. Finally, we show that if $V,V_1,\ldots, V_n$ are valuation domains for a field $K$ and $V[x]\nsubseteq \bigcup_{i=1}^n V_i$ for some $x\in K$, then there exists $v\in V$ such that $v+x\notin \bigcup_{i=1}^n V_i$.
