arXiv 1007.05342010-07-04math.LO

关于$ω$ -范畴群和具有NIP的环

On $ω$-categorical groups and rings with NIP

Krzysztof Krupinski

摘要：

证明了具有NIP的$ω$ -范畴环是有限幂零的.证明了具有NIP和fsg的$ω$ -范畴群是有限幂零群.我们还注意到至少有一个强正则类型的$ω$ -范畴群是阿贝尔群。进一步，我们得到了每个$ω$ -范畴的特征单$p$ -群都有一个无穷的可定义阿贝尔子群.另外假设存在一个非代数的，在$\emptyset$上一般稳定的类型，这样的群是阿贝尔群。

英文摘要：

We show that $ω$-categorical rings with NIP are nilpotent-by-finite. We prove that an $ω$-categorical group with NIP and fsg is nilpotent-by-finite. We also notice that an $ω$-categorical group with at least one strongly regular type is abelian. Moreover, we get that each $ω$-categorical, characteristically simple $p$-group with NIP has an infinite, definable abelian subgroup. Assuming additionally the existence of a non-algebraic, generically stable over $\emptyset$ type, such a group is abelian.

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