  We give a broad study of representation and module theory of Rota-Baxter
algebras. Regular-singular decompositions of Rota-Baxter algebras and
Rota-Baxter modules are obtained under the condition of quasi-idempotency.
Representations of an Rota-Baxter algebra are shown to be equivalent to the
representations of the ring of Rota-Baxter operators whose categorical
properties are obtained and explicit constructions are provided.
Representations from coalgebras are investigated and their algebraic Birkhoff
factorization is given. Representations of Rota-Baxter algebras in the tensor
category context is also formulated.
