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2024 Romania National Olympiad, 2

Let $(\mathbb{K},+, \cdot)$ be a division ring in which $x^2y=yx^2,$ for all $x,y \in \mathbb{K}.$ Prove that $(\mathbb{K},+, \cdot)$ is commutative.

1986 Traian Lălescu, 1.2

Tags: abstract algebra , group theory , klein , division rings , superior algebra

Let $ K $ be the group of Klein. Prove that: [b]a)[/b] There is an unique division ring (up to isomorphism), $ D, $ such that $ (D,+)\cong K. $ [b]b)[/b] There are no division rings $ A $ such that $ (A\setminus\{ 0\} ,+)\cong K. $

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