COMMUTATIVITY THEOREMS FOR RINGS WITH CONSTRAINTS ON COMMUTATORS
Main Article Content
Abstract
Let $R$ be a left (resp. right) $s$-unital ring and $m$ be a positive integer. Suppose that for each $y$ in $R$ there exist $J(t)$, $g(t)$, $h(t)$ in $Z[t]$ such that $x^m[x,y]= g(y)[x,y^2f(y)]h(y)$ (resp. $[x,y]x^m= g(y)[x,y^2f(y)]h(y))$ for all $x$ in $R$. Then $R$ is commutative (and conversely). Finally, the result is extended to the case when the exponent $m$ depends on the choice of $x$ and $y$.
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
References
H. A. S. Abu-Jabal and V. Perie, "Commutativity of s-unital rings through a streb result," Rad. Mat., 7 (1991), 73-92.
M. Ashraf and M. A. Quadri, "On commutativity of associative rings," Bull. Austral. Math. Soc., 38 (1988), 267-271.
M. Ashraf, M. A. Quadri and V. W. Jacob, "On commutativity of right s-unital rings," Riv. Mat Univ. Parma, (to appear).
H. E. Bell, "On some commutativity theorems of Berstein," Arch. Math. (Basel), 24 (1973), 34-38. (5] H. E. Bell, "On power maps and ring commutativity," Canad. Math. Bull., 21 (1978), 399-404.
Y. Hirano, Y. Kobayashi and H. Tominaga, "Some polynomial identities and commutativity of s-unital rings," Math. J. Okayama Univ., 24 (1982), 7-13.
T. P. Kezlan, "A note on commutativity of semi prime PI-rings," Math. laponica, 27 (1982), 267-268.
H. Komatsu and H. Tominaga, "Chacron's condition and commutativity theorems," Math. 1 Okayama Univ., 31 (1989), 101-120.
H. Komatsu and H. Tominaga, "Some commutativity theorems for left s-unital rings," Results in Math., 15 (1989), 335-342.
H. Komatsu, T. Nishinaka and H. Tominaga, "On commutativity of rings," Rad. Mat., 6 (1990), 303-311.
E. Psomopoulos, "Commutativity theorem for rings and groups with constraints on commutators," Jnternat. J. Math. & Math. Sci., 7 (1984), 513-517.
W. Streb "Zur Struktur nichtkommutatlver Ringe," Math. ]. Okayama Univ., 31 (1989), 135-140.