[JadeNB]

You are speaking with a representation theorist, too. It seems to me that we agree on the definition. (I agree that I was writing as if `R` were automatically commutative, after having made a big fuss about the possibility that it wasn't. Incidentally, if you feel insufficiently scarred, you might like to expand your stable of algebras: there is the notion of a coalgebra, which is dual to that of an algebra; and I believe that the usual notion of bialgebra is of a ring equipped with the structure both of an algebra and a coalgebra: https://en.wikipedia.org/wiki/Bialgebra ; but maybe it's different in the world of algebras over non-commutative rings, which is not my speciality. Then among the bialgebras are the Hopf algebras, etc.)