发表机构
The Open University of Israel（以色列开放大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文完整刻画了中心有限除环中 Amitsur-Small 环：仅当除环等于其中心或为实闭域上的 Hamilton 四元数代数时成立，否则二元多项式环中即存在收缩失败的反例。
AI 中文摘要
1978年，Amitsur 和 Small 提出如下问题：对于每个除环 $D$，$D[x_1,\ldots,x_n]$ 的极大左理想是否收缩为更小的多项式子环中的极大左理想。Chapman 和作者证明该答案一般情况下是否定的，并将满足此收缩性质恒成立的 $D$ 称为 Amitsur-Small 环。早期工作表明：Hamilton 的实四元数代数是 Amitsur-Small 的；三次除代数不是 Amitsur-Small 的；二次例子仅限于特定的四元数形式；后续工作排除了奇素数次的循环除代数。我们给出中心有限情形下的完整解决。若 $D$ 在其中心 $F$ 上的维数有限，则 $D$ 是 Amitsur-Small 环当且仅当 $D=F$，或 $F$ 是实闭域且 $D$ 是 Hamilton 四元数代数 $(-1,-1)_F$。在所有其他非交换中心有限情形下，失败已在两个变量中出现：存在极大左理想 $M\subseteq D[x,y]$ 使得 $M\cap D[x]$ 在 $D[x]$ 中不是极大的。
英文摘要
In 1978, Amitsur and Small asked whether, for every division ring $D$, maximal left ideals of $D[x_1,\ldots,x_n]$ contract to maximal left ideals in smaller polynomial subrings. Chapman and the author showed that the answer is negative in general and called $D$ an Amitsur-Small ring when this contraction property always holds. Earlier work showed that Hamilton's real quaternion algebra is Amitsur-Small, that division algebras of degree three are not Amitsur-Small, and that degree-two examples are restricted to a specific quaternionic form; subsequent work excluded cyclic division algebras of odd prime degree. We give a complete resolution in the centrally finite case. If $D$ has finite dimension over its center $F$, then $D$ is an Amitsur-Small ring if and only if either $D=F$, or $F$ is real closed and $D$ is the Hamilton quaternion algebra $(-1,-1)_F$. In every other noncommutative centrally finite case, failure already occurs in two variables: there is a maximal left ideal $M\subseteq D[x,y]$ such that $M\cap D[x]$ is not maximal in $D[x]$.
