Abstract
We address the unital right alternative bimodules over the matrix algebras \( {\mathrm{M}}_{n}(\Phi) \) of order \( n\geq 3 \), prove that each of these bimodules is the direct sum of an associative bimodule and a Graves bimodule, and fully describe the structure of twisted Graves bimodules. Also, we construct an irreducible right alternative \( {\mathrm{M}}_{n}(\Phi) \)-bimodule of minimal dimension \( n(n-1) \). Furthermore, we show that no element \( f(x,y) \) of the free right alternative algebra of rank 3 is its nuclear element. The results of this article are needed for the study of the right alternative superalgebras whose even part includes \( {\mathrm{M}}_{n}(\Phi) \) with \( n\geq 3 \).
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The first author is supported by the FAPESP 2018/23690–6.
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Translated from Sibirskii Matematicheskii Zhurnal, 2022, Vol. 63, No. 4, pp. 893–910. https://doi.org/10.33048/smzh.2022.63.415
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Murakami, L.I., Pchelintsev, S.V. & Shashkov, O.V. Right Alternative Unital Bimodules over the Matrix Algebras of Order \( \geq 3 \). Sib Math J 63, 743–757 (2022). https://doi.org/10.1134/S0037446622040152
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DOI: https://doi.org/10.1134/S0037446622040152