2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/165327We study the ring theoretical structures of mixable shuffle algebras and their associated free commutative Rota-Baxter algebras. For this study we utilize the connection of the mixable shuffle algebras with the overlapping shuffle algebra of Hazewinkel, quasi-shuffle algebras of Hoffman and quasi-symmetric functions. This connection allows us to apply methods and results on shuffle products and Lyndon words on ordered sets. As a result, we obtain structure theorems for a large class of mixable shuffle algebras and free commutative Rota-Baxter algebras with various coefficient rings.30 pages. Corrected typos and improved presentationRings and AlgebrasCommutative AlgebraCombinatoricsNumber Theory16W99, 18F20, 05E99text