Nil subrings of endomorphism rings of finitely generated modules over affine PI-rings.
The authors prove the following result. Let \(R\) be an affine PI-algebra over a commutative Noetherian ring \(C\). Let \(M_R\) be a finitely generated \(R\)-module. Then any nil subring of \(\text{End}_R(M)\) is nilpotent. The authors point out the following two corollaries of this result. Let \(R\) be an affine PI-algebra over a commutative Noetherian ring \(C\), and let \(M_R\) be a finitely generated \(R\)-module. If \(V\) is a nil subset of \(\text{End}_R(M)\) closed under the Lie bracket, then \(V\) generates a nilpotent ring. To formulate the second corollary we recall that a Lie ring \(L\) is said to be weakly Engel if given \(x\) from \(L\) there exists a positive integer \(n(x)\) such that \(\text{ad}(x)^{n(x)}(y)=0\) for any \(y\in L\). The second corollary asserts that for any affine PI-algebra \(A\) over a commutative Noetherian ring \(C\) if \(L\) is a weakly Engel Lie subring of \(A\), then \(L\) is a nilpotent Lie ring. Other corollaries are pointed out.
scientific article; zbMATH DE number 5853360