Polynomial functions on a class of finite non-commutative rings

Amr Ali Abdulkader Al-Maktry, Susan F. El-Deken
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Abstract

Let $R$ be a finite non-commutative ring with $1\ne 0$. By a polynomial function on $R$, we mean a function $F\colon R\longrightarrow R$ induced by a polynomial $f=\sum\limits_{i=0}^{n}a_ix^i\in R[x]$ via right substitution of the variable $x$, i.e. $F(a)=f(a)= \sum\limits_{i=0}^{n}a_ia^i$ for every $a\in R$. In this paper, we study the polynomial functions of the free $R$-algebra with a central basis $\{1,\beta_1,\ldots,\beta_k\}$ ($k\ge 1$) such that $\beta_i\beta_j=0$ for every $1\le i,j\le k$, $R[\beta_1,\ldots,\beta_k]$. %, the ring of dual numbers over $R$ in $k$ variables. Our investigation revolves around assigning a polynomial $\lambda_f(y,z)$ over $R$ in non-commutative variables $y$ and $z$ to each polynomial $f$ in $R[x]$; and describing the polynomial functions on $R[\beta_1,\ldots,\beta_k]$ through the polynomial functions induced on $R$ by polynomials in $R[x]$ and by their assigned polynomials in the in non-commutative variables $y$ and $z$. %and analyzing the resulting polynomial functions on $R[\beta_1,\ldots,\beta_k]$. By extending results from the commutative case to the non-commutative scenario, we demonstrate that several properties and theorems in the commutative case can be generalized to the non-commutative setting with appropriate adjustments.
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一类有限非交换环上的多项式函数
让 $R$ 是一个有 $1\ne 0$ 的有限非交换环。关于 $R$ 上的多项式函数,我们指的是由 R[x]$ 中通过变量 $x$ 的右置换引起的多项式函数 $F\colon R\longrightarrow R$,即对于 R$ 中的每一个 $a/$,函数 $F(a)=f(a)=\sum\limits_{i=0}^{n}a_ia^i$。在本文中,我们研究的是自由 $R$-algebra 的多项式函数,它有一个中心基$\{1,\beta_1,\ldots,\beta_k\}$($k\ge 1$),使得 $\beta_i\beta_j=0$ foreververy $1\le i,j\le k$,$R[\beta_1,\ldots,\beta_k]$。%,是 $k$ 变量中 $R$ 上的对偶数环。我们的研究围绕着在非交换变量 $y$ 和 $z$ 的 $R$ 上为 R[x]$ 中的每个多项式 $f$ 分配一个多项式 $\lambda_f(y,z)$;通过$R[x]$中的多项式及其在非交换变量$y$和$z$中分配的多项式在$R$上引起的多项式函数来描述$R[\beta_1,\ldots,\beta_k]$上的多项式函数。分析在$R[\beta_1,\ldots,\beta_k]$上得到的多项式函数。通过将交换情况下的结果推广到非交换情况下,我们证明交换情况下的一些性质和定理可以通过适当的调整推广到非交换情况下。
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