{"title":"一类有限非交换环上的多项式函数","authors":"Amr Ali Abdulkader Al-Maktry, Susan F. El-Deken","doi":"arxiv-2409.10208","DOIUrl":null,"url":null,"abstract":"Let $R$ be a finite non-commutative ring with $1\\ne 0$. By a polynomial\nfunction on $R$, we mean a function $F\\colon R\\longrightarrow R$ induced by a\npolynomial $f=\\sum\\limits_{i=0}^{n}a_ix^i\\in R[x]$ via right substitution of\nthe variable $x$, i.e. $F(a)=f(a)= \\sum\\limits_{i=0}^{n}a_ia^i$ for every $a\\in R$. In this paper,\nwe study the polynomial functions of the free $R$-algebra with a central basis\n$\\{1,\\beta_1,\\ldots,\\beta_k\\}$ ($k\\ge 1$) such that $\\beta_i\\beta_j=0$ for\nevery $1\\le i,j\\le k$, $R[\\beta_1,\\ldots,\\beta_k]$. %, the ring of dual numbers\nover $R$ in $k$ variables. Our investigation revolves around assigning a polynomial $\\lambda_f(y,z)$\nover $R$ in non-commutative variables $y$ and $z$ to each polynomial $f$ in\n$R[x]$; and describing the polynomial functions on $R[\\beta_1,\\ldots,\\beta_k]$\nthrough the polynomial functions induced on $R$ by polynomials in $R[x]$ and by\ntheir assigned polynomials in the in non-commutative variables $y$ and $z$.\n%and analyzing the resulting polynomial functions on\n$R[\\beta_1,\\ldots,\\beta_k]$. By extending results from the commutative case to the non-commutative\nscenario, we demonstrate that several properties and theorems in the\ncommutative case can be generalized to the non-commutative setting with\nappropriate adjustments.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"75 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Polynomial functions on a class of finite non-commutative rings\",\"authors\":\"Amr Ali Abdulkader Al-Maktry, Susan F. El-Deken\",\"doi\":\"arxiv-2409.10208\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $R$ be a finite non-commutative ring with $1\\\\ne 0$. By a polynomial\\nfunction on $R$, we mean a function $F\\\\colon R\\\\longrightarrow R$ induced by a\\npolynomial $f=\\\\sum\\\\limits_{i=0}^{n}a_ix^i\\\\in R[x]$ via right substitution of\\nthe variable $x$, i.e. $F(a)=f(a)= \\\\sum\\\\limits_{i=0}^{n}a_ia^i$ for every $a\\\\in R$. In this paper,\\nwe study the polynomial functions of the free $R$-algebra with a central basis\\n$\\\\{1,\\\\beta_1,\\\\ldots,\\\\beta_k\\\\}$ ($k\\\\ge 1$) such that $\\\\beta_i\\\\beta_j=0$ for\\nevery $1\\\\le i,j\\\\le k$, $R[\\\\beta_1,\\\\ldots,\\\\beta_k]$. %, the ring of dual numbers\\nover $R$ in $k$ variables. Our investigation revolves around assigning a polynomial $\\\\lambda_f(y,z)$\\nover $R$ in non-commutative variables $y$ and $z$ to each polynomial $f$ in\\n$R[x]$; and describing the polynomial functions on $R[\\\\beta_1,\\\\ldots,\\\\beta_k]$\\nthrough the polynomial functions induced on $R$ by polynomials in $R[x]$ and by\\ntheir assigned polynomials in the in non-commutative variables $y$ and $z$.\\n%and analyzing the resulting polynomial functions on\\n$R[\\\\beta_1,\\\\ldots,\\\\beta_k]$. By extending results from the commutative case to the non-commutative\\nscenario, we demonstrate that several properties and theorems in the\\ncommutative case can be generalized to the non-commutative setting with\\nappropriate adjustments.\",\"PeriodicalId\":501136,\"journal\":{\"name\":\"arXiv - MATH - Rings and Algebras\",\"volume\":\"75 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-16\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Rings and Algebras\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.10208\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Rings and Algebras","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.10208","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Polynomial functions on a class of finite non-commutative rings
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.