A note on commutativity of semiprime PI-rings. (English) Zbl 0481.16013

Let \(f(x_1,x_2,\ldots,x_n)\) be a polynomial in \(n\) noncommuting indeterminates with relatively prime integer coefficients. It is proved that the following three conditions are equivalent:
(1) Every ring satisfying the identity \(f=0\) has nil commutator ideal;
(2) Every semiprime ring satisfying \(f=0\) is commutative;
(3) There exists no prime \(p\) for which the ring of \(2\times 2\) matrices over \(\mathrm{GF}(p)\) satisfies \(f=0\).
This widely-applicable result extends earlier results of the author [Trans. Am. Math. Soc. 125, 414–421 (1966; Zbl 0154.02702)] and the reviewer [Arch. Math. 24, 34–38 (1973; Zbl. 251.16021)].


16U70 Center, normalizer (invariant elements) (associative rings and algebras)
16R99 Rings with polynomial identity
16N60 Prime and semiprime associative rings