On rings which are sums of two PI-subrings: a combinatorial approach.
Let \(R\) be a ring that equals the sum of its subrings \(A\) and \(B\), and suppose that both \(A\) and \(B\) satisfy polynomial identities. An open and quite difficult problem in ring theory is to determine whether \(R\) itself satisfies a polynomial identity or not. Recall that the sum of \(A\) and \(B\) need not be direct; in fact, the interesting situation arises when it is not direct. Up till now the answer to the above problem is known to be positive only under some restrictions on the identities satisfied by either \(A\) or \(B\), or both. The paper under review answers in the positive the question for large classes of rings \(A\) and \(B\). Thus for example the authors prove that if, for some \(k\), \((AB)^k\subseteq A\) or \((BA)^k\subseteq A\) then \(R\) is a PI ring. The same conclusion holds if, for example, \(A\) is a (one-sided) ideal of \(R\), or if \(AB=BA\) and \(B^k\subseteq A\), or if \(A\) is a Lie ideal of \(R\). The authors use a promising technique based on the study of the multilinear polynomial identities and their codimension sequence. This technique allows them to find explicit identities satisfied by \(R\) provided one knows the identities in \(A\) and \(B\). Methods based on the use of multilinear identities and their codimensions have proved extremely efficient in PI theory. Although this paper does not solve the problem in its generality the results obtained seem the strongest and most general in this direction, and could be a first step to solving the above mentioned general problem.
scientific article; zbMATH DE number 2101059