Commutativity theorems for certain rings with polynomial constraints (Q5936398)

This paper presents several theorems asserting commutativity of rings satisfying complicated and artificial commutator constraints. The following result is typical: Let \(p\), \(q\), \(r\) be nonnegative integers with \(r>1\), and let \(R\) be a ring with \(1\) such that \(r[x,y]=0\) implies \([x,y]=0\). If for each \(y\in R\) there exist \(f(t),g(t),h(t)\in\mathbb{Z}[t]\) such that \(g(y)[x,f(y)]h(y)=x^p[x^r,y]y^q\) for all \(x\in R\), then \(R\) is commutative. The proof given for the above theorem does not work for \(p=0\); some other proofs have problems as well.