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Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups. For questions specific to commutative algebra (that is, rings that are assumed both associative and commutative), rather use the tag ac.commutative-algebra.
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Finite dimensional real division algebras
If you ask for a purely algebraic proof, algebraic K-theory is (almost) OK. Indeed for a compact Hausdorff space $X$, it's topological K-theory $K(X)$ is the same as algebraic K-theory $K_0(C(X;\math …