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## Scientific Bulletin. Physical and Mathematical Research

#### Abstract

Associative algebras are introduced into mathematics of the 19th century and are still intensively studied. The classification of associative algebras of small dimensions first appeared in the works of Pierce in 1881. In 2018, the German scientist William de Graf gave a classification of nilpotent associative algebras of small sizes. The article describes all the Rota-Baxter operators on 3-dimensional nilpotent associative algebras.

Rota-Baxter operators were defined by Baxter to solve an analytic formula in probability. It has been related to other areas in mathematical physics and mathematics.

Throughout this paper algebras are considered over the field of complex numbers.

A Rota-Baxter operator on an associative algebra A over a field F is defined to be a linear map P : A → A satisfying

P(x)P(y) = P(xP(y) + P(x)y + λxy), ∀x,y ∈ A, λ ∈ F.

Note that, if P is a Rota-Baxter operator of weight λ≠0, then λ−1P is a Rota-Baxter operator of weight 1. Therefore, it is sufficient to consider Rota-Baxter operators of weight 0 and 1.

Any three-dimensional complex nilpotent associative algebra A is isomorphic to one of the five pairwise non-isomorphic algebras named by A1, A2, A3, A4 and A5 which presented above.

We proof that there are five types for 3-dimensional associative algebra A1, three types for A2 and A3, two types for A5 of Rota-Baxter operators of weight 0. Moreover, we have showed there are eight types for 3-dimensional associative algebra A1, one type for A2, four types for A3, one type for A4 and two types for A5 of Rota-Baxter operators of weight 1.

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