Azumaya-like properties of verbally prime algebras
This paper continues the investigations on the verbally prime algebras introduced by \textit{A. R. Kemer} [Izv. Akad. Nauk SSSR, Ser. Mat. 48, No.5, 1042-1059 (1984; Zbl 0586.16010)]. Assume A is a verbally prime PI algebra over a field of characteristic 0 and Z is the centre of A. The main theorem in the paper states that there exists a non-nilpotent ideal M of Z satisfying the following conditions. 1. If \(c\in M\) then cA is contained in a finitely generated Z-module. 2. If \(J\triangleleft A\) then MJ\(\subset (J\cap Z)A\subset J\). 3. If \(J\triangleleft Z\) then M(JA\(\cap Z)\subset J\subset (JA\cap Z)\). 4. Assume \(\theta\) : \(A\otimes_ ZA^{op}\to End_ Z(A)\) is the canonical map. Then M Ker \(\theta\) \(=0\) and M End\({}_ Z(A)\subset Im \theta\). The proof uses the result quoted of Kemer and of \textit{Yu. P. Razmyslov} [Mat. Sb., Nov. Ser. 128, No.2, 194-215 (1985; Zbl 0601.16016)].
scientific article; zbMATH DE number 4158879