Modules of Non-unital Polynomial Rota-Baxter Algebras

In this paper, we study the free commutative non-unital Rota-Baxter algebra which is the algebra of polynomials in one variable without constant term with Rota-Baxter operators of nonzero weight. The main result shows that every module over this Rota-Baxter algebra is equivalent to the modules over a noncommutative algebra generated by two indeterminates with a generation relationship. Furthermore, we study their irreducible and indecomposable modules. Finally we provide the classification of modules of nonzero weight through solution to some matrix equations.