"Let R be a commutative ring. We say that M is an(adsbygoogle = window.adsbygoogle || []).push({}); algebra overR, or that M is anR-algebraif M is an R-module that is also a ring (not necessarily commutative), and the ring and module operations are compatible, i.e., [tex]r(xy) = (rx)y = x(ry)[/tex] for all [itex]x, y \in M[/itex] and [itex]r \in R[/itex]."

I'm not really sure why the second equality is true, because it implies commutativity and the definition tells us that an R-module is not necessarily commutative, right?

Thank you in advance

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# The definition of an algebra

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