An independence property of central polynomials (Q1175125)

Let \(\Phi\) be a field and \(X=\{x_ 1,x_ 2,\dots,x_ n,\dots\}\) be a set of noncommuting indeterminates over \(\Phi\). Let \(\Phi\{X\}\) be the free algebra generated by \(X\). Theorem. Let \(I_ n=\{f\in\Phi\{X\}\): \(f\) is a polynomial identity of \(M_ n(\Phi)\}\) be the \(T\)-ideal of polynomial identities of \(M_ n(\Phi)\). Let \(f_ i(x_ 1,\dots,x_ r)\) and \(g_ i(x_{r+1},\dots,x_ s)\), \(1\leq i\leq k\) be polynomials in \(\Phi\{X\}\) with disjoint indeterminates. Assume that \(\sum_{i=1}^ k f_ i(x_ 1,\dots,x_ r)g_ i(x_{r+1},\dots,x_ s)\) is a central polynomial on \(M_ n(\Phi)\). Then except when \(k\geq 2\), \(n=2\) and \(\Phi=\mathbb{Z}/(2)\) the following hold: (1) If \(f_ i\), \(i=1,2,\dots,k\) are \(\Phi\)-independent modulo \(I_ n\) then \(g_ i\) must be central \(\forall i\). (2) If \(g_ i\), \(i=1,2,\dots,k\) are \(\Phi\)-independent modulo \(I_ n\) then \(f_ i\) must be central \(\forall i\). This generalizes a theorem of \textit{A. Regev} [Pac. J. Math. 83, 269-271 (1979; Zbl 0425.16017)]. Examples are given to show that the result is false when \(k\geq 2\), \(n=2\) and \(\Phi=\mathbb{Z}/(2)\).