Abstract
We describe the right-symmetric algebras of matrix type \( M_{2}(F) \) over a field \( F \) of characteristic \( 0 \) such that the left action of the orthogonal idempotents of \( M_{2}(F) \) is diagonalizable, and the right-module part \( W \) includes no constant bichains. We construct some wide class of nonassociative algebras \( E_{\psi,\partial}(W,{\mathcal{A}}) \), where \( W \) is a subalgebra and a right module over an associative algebra \( {\mathcal{A}} \). We give a criterion for these algebras to be right-symmetric. Assuming that \( W{\mathcal{A}}=W \), we show that the algebras of this class are either simple or local. We exhibit some examples of simple right-symmetric algebras and right-symmetric algebras without nilpotent right ideals whose right-module part is not an irreducible module over \( M_{2}(F) \).
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Acknowledgment
The author is deeply grateful to the referee for many valuable remarks that improve the article.
Funding
This research was supported by the Russian Science Foundation (Project 21–11–00286).
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Translated from Sibirskii Matematicheskii Zhurnal, 2023, Vol. 64, No. 4, pp. 773–785. https://doi.org/10.33048/smzh.2023.64.410
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Pozhidaev, A.P. On Diagonal Nonconstant Right-Symmetric Algebras of Matrix Type \( M_{2}(F) \). Sib Math J 64, 879–889 (2023). https://doi.org/10.1134/S0037446623040109
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DOI: https://doi.org/10.1134/S0037446623040109