AI 中文总结
本文研究极小非交换环，将有限情形的这类环分为三类并完成前两类刻画，给出第三类的有限生成过程，还将问题转化为交换代数问题，指出无限极小非交换环若存在则为反例。
AI 中文摘要
我们研究极小非交换环，即所有真子环和同态像均为交换环的非交换环。这类环由Bell和Danchev引入，用于验证交换性定理，他们提出了有限和无限情形下所有这类环的描述问题。在有限情形，我们将其划分为三个互不相交的类，其中前两类已完全刻画；对第三类，我们给出了一个有限过程，可生成所有所需的这类环且仅能生成这类环。我们还将该问题转化为交换代数中具有小基座（socle）和一类消去性质的有限局部环问题。最后，我们证明若存在无限极小非交换环，则它是具有非常奇特性质的除环，且是若干长期猜想的反例。
英文摘要
We study minimal noncommutative rings, that is noncommutative rings whose proper subrings and homomorphic images are all commutative. These rings were introduced by Bell and Danchev in order to test commutativity theorems. They raised the problems of describing all such rings in the finite and infinite cases. In the finite case, we give a classification into three pairwise disjoint classes, the first two of which are completely characterised. For the third class, we give a finite procedure which can produce any of (and only) the required rings. We also translate the problem into commutative algebra, in terms of finite local rings with small socle and a kind of cancellation property. Finally, we show that if an infinite minimal noncommutative ring exists, then it is a division algebra with very strange properties, and a counterexample to several longstanding conjectures.
Comments31 pages
