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YCor
YCor
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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})_{ab}$$(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})_{ab}\to (R/I)^{\times}$$(R^{\times})_{\mathrm{ab}}\to (R/I)^{\times}$ but this map does not have to be an isomorphism).

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 $(R^{\times})_{ab}$?

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

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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Ehud Meir
Ehud Meir
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Abelianization of the group of invertible elements in a finite local ring

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 $(R^{\times})_{ab}$?

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