类型较少的域和环
Fields and rings with few types
摘要：
设R是具有可能额外结构的结合环。称R是弱小的，如果在R的任何有限子集上有可数多个1-型。它是局部P，如果R的任何有限子集的代数闭包具有性质P。本文证明了弱小域的有限次域扩张要么是有限域，要么没有Artin-Schreier扩张。特征为2的弱小域是有限的或代数闭的。每个正特征的弱小除环在其中心上是局部有限维的。弱小环的Jacobson根是局部幂零的。每一个弱小除环局部模其Jacobson根同构于除环上的n个矩阵环的乘积。
英文摘要：
Let R be an associative ring with possible extra structure. R is said to be weakly small if there are countably many 1-types over any finite subset of R. It is locally P if the algebraic closure of any finite subset of R has property P. It is shown here that a field extension of finite degree of a weakly small field either is a finite field or has no Artin-Schreier extension. A weakly small field of characteristic 2 is finite or algebraically closed. Every weakly small division ring of positive characteristic is locally finite dimensional over its centre. The Jacobson radical of a weakly small ring is locally nilpotent. Every weakly small division ring is locally, modulo its Jacobson radical, isomorphic to a product of finitely many matrix rings over division rings.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
