论文标题
在小代数的右核上
On the Right Nucleus of Petit Algebras
论文作者
论文摘要
让$ d $为中心$ c $的分区代数，让$σ$是$ d $的增强型，让$δ$是$σ$ deb的$ d $，而让$ r = d [t;σ，δ] $为偏斜的多元式戒指。我们研究了一类非缔合代数的结构，该代数为$ s_f $，其构造在规范上概括了协会商代数$ r/rf $，其中r $中的$ f \ in R $是直接不变的。 当多项式$ f $有限且不变时，$δ= 0 $或$σ= {\ rm ID} _d $时，我们确定$ s_f $的正确核的结构。作为副产品，我们获得了有关循环（Petit）semifield $ \ mathbb {s} _f $的右核大小的新证明。 我们查看$ s_f $的右核的子代理，概括了Petit的几个结果\ cite {petit1966-clines}，并介绍了系数环$ d $的半不变元素的概念。当$ f $不是正确的不变时，一组半不变元素等于$ s_f $的核。此外，我们计算某些$ f $的$ s_f $的正确核。 在本文的最后一章中，我们介绍和研究了一类特殊的多项式类别，称为概括性a-polynomials。在特征零字段上的差分多项式环中，Amitsur \ cite {amitsur1954differential}最初引入了A-PolyNomials。我们找到了多项式的示例，其本征是在字段上的中央简单代数$ c \ cap {\ rm fix}（σ）\ cap {\ rm const}（δ）$。
Let $D$ be division algebra over its center $C$, let $σ$ be an endormorphism of $D$, let $δ$ be a left $σ$-derivation of $D$, and let $R=D[t;σ,δ]$ be a skew polynomial ring. We study the structure of a class of nonassociative algebras, denoted by $S_f$, whose construction canonically generalises that of the associative quotient algebras $R/Rf$ where $f\in R$ is right-invariant. We determine the structure of the right nucleus of $S_f$ when the polynomial $f$ is bounded and not right invariant and either $δ= 0$, or $σ= {\rm id}_D$. As a by-product, we obtain a new proof on the size of the right nuclei of the cyclic (Petit) semifields $\mathbb{S}_f$. We look at subalgebras of the right nucleus of $S_f$, generalising several of Petit's results \cite{petit1966certains} and introduce the notion of semi-invariant elements of the coefficient ring $D$. The set of semi-invariant elements is shown to be equal to the nucleus of $S_f$ when $f$ is not right-invariant. Moreover, we compute the right nucleus of $S_f$ for certain $f$. In the final chapter of this thesis we introduce and study a special class of polynomials in $R$ called generalised A-polynomials. In a differential polynomial ring over a field of characteristic zero, A-polynomials were originally introduced by Amitsur \cite{amitsur1954differential}. We find examples of polynomials whose eigenring is a central simple algebra over the field $C \cap {\rm Fix}(σ) \cap {\rm Const}(δ)$.
