A central quaternionic Nullstellensatz
Let \(\mathbb{H}\) be the quaternion algebra and \(R=\mathbb{H}[x_1,\dots,x_n]\) the ring of polynomials over \(\mathbb{H}\) in \(n\) central variables \(x_1,\dots,x_n\). Denote by \(\mathbb{H}^n_c\) the set of \((a_1,\dots,a_n)\in \mathbb{H}^n\) such that \(a_ia_j=a_ja_i\) for all \(i,j\). The first main result of the paper is the following. Theorem (weak Nullstellensatz). For any proper left ideal \(I\) in \(R\) there exists \((a_1,\dots,a_n)\in \mathbb{H}^n_c\) such that \(p(a_1,\dots,a_n)=0\) for all \(p\in I\). Given a left ideal \(I\) in \(R\), denote by \(Z(I)\) the set of \((a_1,\dots,a_n) \in \mathbb{H}^n_c\) such that \(p(a_1,\dots,a_n)=0\) for all \(p\in I\). For any subset \(Z\subseteq \mathbb{H}^n_c\) denote by \(I(Z)\) the left ideal of \(p\in R\) such that \(p(a_1,\dots,a_n)=0\) for all \((a_1,\dots,a_n) \in \mathbb{H}^n\). A left ideal \(I\) in \(R\) is called \textit{completely prime} if for all \(a, b \in R\) \[ab \in I\text{ and }Ib \subseteq I\ \Rightarrow\ a \in I\text{ or }b \in I. \] Let \(\sqrt I\) denote the intersection of all prime ideals that contain \(I\). The second main result of the paper is the following. Theorem (strong Nullstellensatz). Let \(I\) be a proper left ideal in \(R\). Then \(I(Z(I))=\sqrt I\).
scientific article; zbMATH DE number 7318816
scientific article