An additivity principle for p.i. rings
Let \({\mathcal R}\) be a prime ring with polynomial identity, \({\mathcal Z}\) is the center of \({\mathcal R}\) and \({\mathcal T}({\mathcal R})\) is the trace ring which is generated by the coefficients of the characteristic polynomials of elements of \({\mathcal R}\) over \({\mathcal Z}\). Let also \(p\in Spec {\mathcal Z}\), \({\mathcal I}=\{{\mathcal Q}\); \({\mathcal Q}\in Spec {\mathcal R}\), \({\mathcal Q}\cap {\mathcal Z}=p\}\), \(| {\mathcal I}| =s\). The author proves that if \({\mathcal T}({\mathcal R})={\mathcal R}\) then p.i.d.(\({\mathcal R})=\sum^{s}_{i=1}K_ i\cdot (p.i.d.({\mathcal R}/{\mathcal P}_ i))\), where \({\mathcal P}_ i\in {\mathcal I}\), \(k_ i\geq [{\mathcal Z}({\mathcal R}_ p/{\mathcal P}_{ip})_{sep}:{\mathcal Z}_ p/{\mathcal P}_ p]\), \(i=1,2,...,s\). As a corollary the author gives a new proof of the famous Bergman-Small theorem.
scientific article; zbMATH DE number 3920713
scientific article