International Journal of Mathematics Trends and Technology
Volume 66 | Issue 10 | Year 2020 | Article Id. IJMTT-V66I10P504 | DOI : https://doi.org/10.14445/22315373/IJMTT-V66I10P504
H.E. Bell[2] proved that an n-torsion free ring with identity which satisfies the identity(xy)n = (yx)n is necessarily commutative. More recently, Y. Hirano, M. Hongan and H.Tominaga has proved that same for s-unital rings[5]. On the other hand, H. Abu-Khuzam paper has proved that an (n+1)n-torsion free ring with identity which satisfies the identity(xy)n+1 = xn+1 yn+1 is commutative[1]. Our objective is to generalize Bell's result to s-unital rings satisfying weaker identities which are implied by the identity(xy)n=(yx)n and to generalize the main theorem of [1] to s-unital rings. As stated in [5], if R is an s-unital ring, then for any finite subset F of R, there exists an element e in R such that ex = xe = e for all x in F. Such an element e will be called a pseudo-identity of F. Throughout, R will respresent a ring with ceter C, and N will denote the set of all nilpotent elements of R. As usual, [x,y] will denote the commutator xy-yx. Our aim is to prove the following theorems.
[1] H. ABU-KHUZAM : A commutativity theorem for rings, Math. Japonica, to appear.
[2] H.E. BELL : On rings with commuting powers, Math. Japonica 24 (1979), 473—478.
[3] I.N. HERSTEIN : Power maps in rings, Michigan Math. J. 8 (1961), 29—32.
[4] I.N. HERSTEIN : A commutativity theorem, J. Algebra 38 (1976), 112—118.
[5] Y. HIRANO, M. HONGAN and H. TOMINAGA : Commutativity theorems for certain rings, Math. J. Okayama Univ. 22 (1980), 65-72
S.Lalitha, Dr.S. Sreenivasulu, Prof. A. Mallikarjun Reddy, "Some Commutativity Results For s-Unital Rings Satisfying Polynomial Identities," International Journal of Mathematics Trends and Technology (IJMTT), vol. 66, no. 10, pp. 20-23, 2020. Crossref, https://doi.org/10.14445/22315373/IJMTT-V66I10P504
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