On $ω$-categorical groups and rings with NIP

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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