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This paper presents a cohomological study of modified Rota-Baxter associative algebras in the presence of derivations. The Modified Rota-Baxter operator, which is a modified version and closely related to the classical Rota-Baxter operator, has garnered significant attention due to its applications in various mathematical and physical contexts. In this study, we define a cohomology theory and also investigate a one-parameter formal deformation theory and abelian extensions of modified Rota-Baxter associative algebras under the influence of derivations.
Basdouri et al. (Sun,) studied this question.