  A Rota--Baxter operator is an algebraic abstraction of integration, which is
the typical example of a weight zero Rota-Baxter operator. We show that
studying the modules over the polynomial Rota--Baxter algebra $(k[x],P)$ is
equivalent to studying the modules over the Jordan plane, and we generalize the
direct decomposability results for the $(k[x],P)$-modules in [Iy] from
algebraically closed fields of characteristic zero to fields of characteristic
zero. Furthermore, we provide a classification of Rota--Baxter modules up to
isomorphism based on indecomposable $k[x]$-modules.
