Commutativity of rings through a Streb’s result. (English) Zbl 1079.16504

Summary: We investigate commutativity of rings with unity satisfying any one of the properties: \[ \begin{aligned} &\{1-g(yx^m)\}[yx^m-x^rf(yx^m)x^s,x]\{1-h(yx^m)\}=0,\\ &\{1-g(yx^m)\}[x^my-x^rf(yx^m)x^s,x]\{1-h(yx^m)\}=0,\\ &y^t[x,y^n]=g(x)[f(x),y]h(x)\quad\text{and}\quad [x,y^n]y^t=g(x)[f(x),y]h(x)\end{aligned} \] for some \(f(X)\) in \(X^2\mathbb{Z}[X]\) and \(g(X),h(X)\) in \(\mathbb{Z}[X]\), where \(m\geq 0\), \(r\geq 0\), \(s\geq 0\), \(n>0\), \(t>0\) are non-negative integers. We also extend these results to the case when integral exponents in the underlying conditions are no longer fixed, rather they depend on the pair of ring elements \(x\) and \(y\) for their values. Further, under different appropriate constraints on commutators, commutativity of rings is studied. These results generalize a number of commutativity theorems established recently.


16U70 Center, normalizer (invariant elements) (associative rings and algebras)
16R50 Other kinds of identities (generalized polynomial, rational, involution)
16U80 Generalizations of commutativity (associative rings and algebras)
Full Text: DOI EuDML


[1] H. E. Bell, M. A. Quadri and M. A. Khan: Two commutativity theorems for rings. Rad. Mat. 3 (1987), 255-260. · Zbl 0648.16028
[2] M. Chacron: A commutativity theorem for rings. Proc. Amer. Math. Soc. 59 (1976), 211-216. · Zbl 0341.16020
[3] I. N. Herstein: Two remarks on commutativity of rings. Canad. J. Math. 7 (1955), 411-412. · Zbl 0065.02203
[4] T. P. Kezlan: A note on commutativity of semiprime PI-rings. Math. Japon. 27 (1982)), 267-268. · Zbl 0481.16013
[5] M. A. Khan: Commutativity of right \(s\)-unital rings with polynomial constraints. J. Inst. Math. Comput. Sci. 12 (1999), 47-51. · Zbl 0935.16023
[6] H. Komatsu and H. Tominaga: Chacron’s condition and commutativity theorems. Math. J. Okayama Univ. 31 (1989), 101-120. · Zbl 0705.16023
[7] E. Psomopoulos: Commutativity theorems for rings and groups with constraints on commutators. Internat. J. Math. Math. Sci. 7 (1984), 513-517. · Zbl 0561.16013
[8] M. O. Searoid and D. MacHale: Two elementary generalisations of Boolean rings. Amer. Math. Monthly, 93 (1986), 121-122. · Zbl 0601.16025
[9] W. Streb: Zur Struktur nichtkommutativer Ringe. Math. J. Okayama Univ. 31 (1989), 135-140. · Zbl 0702.16022
[10] H. Tominaga and A. Yaqub: Commutativity theorems for rings with constraints involving a commutative subset. Results Math. 11 (1987), 186-192. · Zbl 0618.16029
[11] J. Tong: On the commutativity of a ring with identity. Canad. Math. Bull. 72 (1984), 456-460. · Zbl 0545.16015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.