One generator algebras

Abstract

For $R_1,R_2,R_3,\dots$ a family of non isomorphic rings (or algebras) having each only 2 idempotents ($1$ and $0$), we classify up to isomorphism the rings (or algebras) obtained by taking products of powers of the different $R_i$. We show that the automorphism groups of such rings (or algebras) split naturally into the product of wreath products $Aut( R_n)\wr \mathfrak{S}_{m_n} $ for different $n$. These results are applied to algebras generated by one element over a perfect field $\mathbb{K}$. Such algebra is either $\mathbb{K}[X]$ or a quotient of $\mathbb{K}[X]$. We show that in the later case the algebra is isomorphic to a finite product of the form $A=\prod (\mathbb{L}_i[X]/(X^j))^{n_{i,j}}$, where the $\mathbb{L}_i$ are non isomomorphic finite field extensions of $\mathbb{K}$ $($not isomophic as $\mathbb{K}$-algebras$)$, with restrictions on the numbers $n_{i,j}$ if $\mathbb{K}$ is finite. We classify these algebras up to isomorphism. We have also that the $\mathbb{K}$-algebra automorphism group of $A=\prod (\mathbb{L}_i[X]/(X^j))^{n_{i,j}}$ splits naturally into the product of wreat products $Aut_\mathbb{K}(\mathbb{L}_i[X]/(X^j) )\wr \mathfrak{S}_{n_{i,j}}$ ($Aut_\mathbb{K}(-)$ is for $\mathbb{K}$-algebra automorphism group). Finally, we prove that $Aut_\mathbb{K}(\mathbb{L}_i[X]/(X^n) )$ is isomorphic to the semi-direct product $G_n(\mathbb{L}_i)\rtimes Aut_\mathbb{K}(\mathbb{L}_i)$ ($Aut_\mathbb{K}(-)$ is for $\mathbb{K}$-algebra automorphism group), where $G_n(\mathbb{L}_i)\simeq Aut_{\mathbb{L}_i}(\mathbb{L}_i[X]/(X^n) )$ ($\mathbb{L}_i$ algebra automorphism group) is an algebraic subgroup of invertible lower triangular matrices of dimension $(n-1)\times (n-1)$ with coefficients in $\mathbb{L}_i$; the conjugate of a matrix $M\in G_n(\mathbb{L}_i)$ by $\sigma \in Aut_\mathbb{K}(\mathbb{L}_i)$ is the matrix obtained from $M$ by applying $\sigma$ to its coefficients.