  In this paper, we begin a systematic study of modified Rota-Baxter algebras,
as an associative analogue of the modified classical Yang-Baxter equation. We
construct free commutative modified Rota-Baxter algebras by a variation of the
shuffle product and describe the structure both recursively and explicitly. We
then provide these algebras with a Hopf algebra structure by applying a
Hochschild cocycle.
