Gröbner–Shirshov bases for Rota–Baxter algebras

We establish the composition-diamond lemma for associative nonunitary Rota–Baxter algebras of weight $\lambda$. To give an application, we construct a linear basis for a free commutative and nonunitary Rota–Baxter algebra, show that every countably generated Rota–Baxter algebra of weight 0 can be embedded into a two-generated Rota–Baxter algebra, and prove the 1-PBW theorems for dendriform dialgebras and trialgebras.