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Let $R$ be a finite local $\mathbb{F}_q$-algebra. Assume that $R\cong R^*$ as left $R$-modules. Are there any known results about the abelianization $(R^{\times})_{\mathrm{ab}}$?

(We can factor $R$ be the ideal $I$ generated by all additive commutators $xy-yx$. We then get a map $(R^{\times})_{\mathrm{ab}}\to (R/I)^{\times}$ but this map does not have to be an isomorphism).

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  • $\begingroup$ Just to check notation, is $R^*$ the dual $\operatorname{Hom}(R, \mathbb F_q)$? $\endgroup$
    – LSpice
    Commented Sep 16, 2022 at 21:14
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    $\begingroup$ Yes indeed. I know that the socle of $R$ as a left $R$-module is one dimensional, and that enables me to derive such an isomorphism. $\endgroup$
    – Ehud Meir
    Commented Sep 17, 2022 at 7:07

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