A ring which is a sum of two PI subrings is always a PI ring
In 1962 the following problem was posed by \textit{O. H. Kegel} [Math. Ann. 149, 258--260 (1963; Zbl 0106.25402)]: If a ring \(R\) is a sum of two subrings \(R_1\) and \(R_2\) and both \(R_1\) and \(R_2\) satisfy polynomial identities, is then \(R\) a PI ring? (In fact Kegel originally posed it in a somewhat different form but the essence is the same.) This problem attracted the attention of various algebraists, and there have been several partial results. The paper under review settles the problem in its generality. Namely, the author proves that if \(R\) is an associative ring (not necessarily with 1), and \(R=R_1+R_2\) is a sum of two subrings, then \(R\) is PI whenever both \(R_1\) and \(R_2\) are PI. In order to obtain this important result the author makes use of a theorem proved in [the author and \textit{E. R. Puczyłowski}, Commun. Algebra 29, No. 5, 2059--2065 (2001; Zbl 0993.16017)]. The theorem is the following. Let \(T\) be a class of rings that is closed under direct powers and under homomorphisms. If each nontrivial prime ring in \(T\) contains a nontrivial one-sided ideal which is PI as a ring then \(T\) consists of PI rings. It is worth mentioning two interesting corollaries to the main theorem of the paper. {\parindent=6mm \begin{itemize}\item[1.] Let \(R\) be a sum of two nil and PI subrings \(R_1\) and \(R_2\). Then \(R\) is also nil (and PI). \item[2.] Let \(R=R_1+R_2+R_3\) be a ring that is a sum of three PI subrings. If at least one of these subrings is one-sided ideal in \(R\) then \(R\) is PI. \end{itemize}}
scientific article; zbMATH DE number 6808580