The study of free Baxter algebras was started by Rota and Cartier 30 years ago. We continue this study by applying two recent constructions of free Baxter algebras. We investigate the basic structure of a free Baxter algebra and characterize in detail when a free Baxter algebra is a domain or a reduced algebra. We also describe the nilpotent radical of a free Baxter algebra when it is not reduced.
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