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Let $A$ be an algebra over a commutative ring $k$. We prove that braidings onthe category of $A$-bimodules are in bijective correspondence to canonicalR-matrices, these are elements in $A\ot A\ot A$ satisfying certain axioms. Weshow that all braidings are symmetries. If $A$ is commutative, then thereexists a braiding on ${}_A\Mm_A$ if and only if $k\to A$ is an epimorphism inthe category of rings, and then the corresponding $R$-matrix is trivial. If theinvariants functor $G = (-)^A:\{}_A\Mm_A\to \Mm_k$ is separable, then $A$admits a canonical R-matrix; in particular, any Azumaya algebra admits acanonical R-matrix. Working over a field, we find a remarkable newcharacterization of central simple algebras: these are precisely the finitedimensional algebras that admit a canonical R-matrix. Canonical R-matrices giverise to a new class of examples of simultaneous solutions for the quantumYang-Baxter equation and the braid equation.

Braidings on the category of bimodules, Azumaya algebras and epimorphisms of rings