Polynomial functions on a class of finite non-commutative rings
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| Format: | Preprint |
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2024
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| author | Al-Maktry, Amr Ali Abdulkader El-Deken, Susan F. |
| author_facet | Al-Maktry, Amr Ali Abdulkader El-Deken, Susan F. |
| contents | 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 $\{1,β_1,\ldots,β_k\}$ ($k\ge 1$) such that $β_iβ_j=0$ for every $1\le i,j\le k$, $R[β_1,\ldots,β_k]$. %, the ring of dual numbers over $R$ in $k$ variables.
Our investigation revolves around assigning a polynomial $λ_f(y,z)$ over $R$ in non-commutating variables $y$ and $z$ to each polynomial $f$ in $R[x]$; and describing the polynomial functions on $R[β_1,\ldots,β_k]$ through the polynomial functions induced on $R$ by polynomials in $R[x]$ and by their assigned polynomials in the non-commutating variables $y$ and $z$. %and analyzing the resulting polynomial functions on $R[β_1,\ldots,β_k]$.
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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_10208 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Polynomial functions on a class of finite non-commutative rings Al-Maktry, Amr Ali Abdulkader El-Deken, Susan F. Rings and Algebras Combinatorics 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 $\{1,β_1,\ldots,β_k\}$ ($k\ge 1$) such that $β_iβ_j=0$ for every $1\le i,j\le k$, $R[β_1,\ldots,β_k]$. %, the ring of dual numbers over $R$ in $k$ variables. Our investigation revolves around assigning a polynomial $λ_f(y,z)$ over $R$ in non-commutating variables $y$ and $z$ to each polynomial $f$ in $R[x]$; and describing the polynomial functions on $R[β_1,\ldots,β_k]$ through the polynomial functions induced on $R$ by polynomials in $R[x]$ and by their assigned polynomials in the non-commutating variables $y$ and $z$. %and analyzing the resulting polynomial functions on $R[β_1,\ldots,β_k]$. 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. |
| title | Polynomial functions on a class of finite non-commutative rings |
| topic | Rings and Algebras Combinatorics |
| url | https://arxiv.org/abs/2409.10208 |