Monoidal structure on Rota-Baxter algebras via augmentation, and free cocycle bialgebras

A tensor product is known for commutative Rota--Baxter algebras through a universal construction, but its underlying algebra need not be the ordinary tensor product $A\otimes B$ and can be substantially larger. We show that, after passing to augmented Rota--Baxter algebras, the ordinary tensor produ