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Multiple Rota-Baxter algebra and multiple Rota-Baxter modules
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abstract
In this paper, we develop the theory of multiple Rota-Baxter modules over multiple Rota-Baxter algebras. We introduce left, right, and bimodule structures and construct free $\Omega$-operated modules with mixable tensor establishing free commutative multiple Rota-Baxter modules. We provide a necessary and sufficient condition for a free module to admit a free multiple Rota-Baxter module structure. Furthermore, we define projective and injective multiple Rota-Baxter modules, showing that their category has enough projective and injective objects to support derived $\mathrm{Hom}$ functors. Finally, we introduce the tensor product of multiple Rota-Baxter algebras and define flat multiple Rota-Baxter modules, proving that both free and projective modules satisfy the flatness property.
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Nijenhuis modules and the ring of Nijenhuis operators
Nijenhuis modules are shown to be the same as modules over a constructed ring U_N(A), and the category is claimed to have enough projective, injective, and flat objects.
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