Commutativity of rings satisfying certain polynomial identities

Abstract

It is shown that an n-torsion-free ring R with identity such that, for all x, y in R, xnyn = ynyn and (xy)n+1 − xn+1yn+1 is central, must be commutative. It is also shown that a periodic n–torsion-free ring (not necessarily with identity) for which (xy)n − (yx)n is always in the centre is commutative provided that the nilpotents of R form a commutative set. Further, examples are given which show that all the hypotheses of both theorems are essential.

Commutativity of Rings with Constraints Involving a Subset

“…At the end of the sections counterexamples are given which show that the hypotheses are not altogether superfluous. Our theorems generalise the results obtained in [1], [3], [4], [6], [7], [10], [14].…”

“…In a recent paper [1], the author jointly with Abujabal, Bell and Khan proved that R is commutative if R satisfies C 5 (m, R). In their paper [4], Abu-Khuzam et al established commutativity of the m-torsion-free ring R with identity 1 satisfying C 1 (m, R) and C 3 (m + 1, R). Motivated by these observations, it is natural to ask a question: What can we say about the commutativity of R if the property C 3 (m + 1, R) in the above result is replaced by C 5 (m + 1, R)?…”

Commutativity of rings satisfying some polynomial conditions

“…We generalize this theorem by deleting the condition (xy) TM -yn+lxn+l E E C and establishing the commutativity of R under the weaker conditions. We also generalize Theorem A below which was proved in [3]. …”

“…EXAMPLE 4. The ring of all 3 • 3 strictly upper-triangular matrices over GF (3), and where n = 4, shows that the hypothesis "1 E R" cannot be deleted from Theorems 1 and 2.…”