On commutativity of rings with constraints involving a nil subset
DOI:
https://doi.org/10.22199/S07160917.1996.0001.00005Keywords:
Comrnutativity, Ring with identity, S-unítal ringsAbstract
The main theorem of this paper is that a ring R with unity is commutative if and only if there is a nil subset B of R such that
l. for each x ? R, either x ? Z(R) or there is a polynormial f over Z with x - x2f (x) ? B;
2. for each x, y x ? R, there are non-negative integers n > 1, m, r, s depending on a pair of ring elements x,y with x(xmy ± xrynxs) - (xmy ± xrynxs)x = 0.
A related result for a nil commutative subset of R is given and the restrictions on the hypothesis of our result are justified by examples.
References
[1] H. A. S. Abujabal and M. A. Khan, “Commutativity of one sided s-unital rings," Internat. J. Math. and Math. Sci., 15 , pp. 815-818, 1992.
[2] H. A. S. Abujabal, M. A. Khan and M. S. Samman, “On commutativity of one-sided s-unital rings," Tamkang J. Math., 23 , pp. 253- 258, 1992.
[3] H. E. Bell, "On two commutativity properties for rings," Mat . .Japonica, 26 , pp. 523- 528, 1981.
[4] I. N. Herstein, " The structure of certain class of rings," Amer. J. Math., 75 , pp. 864- 871, 1953.
[5] Y. Hirano, Y. Kobayashi and H. Tominaga, '' Some polynomial identities and commutativity of s-unital rings," Math. J. Okayama University, 24 , pp. 7- 13, 1982.
[6] Y. Hirano, M. Hongan and H. Tominaga, "Commutativity theorems for certain rings," Math. J. Okayama University, 22 , pp. 65 - 72, 1980.
[7] I. Mogami and M. Hongan, "Note on commutativity of rings,'' Math. J. Okayama University, :20 , pp. 21 - 24, 1978.
[8] H. Tominaga, "On s-unital rings," Math. J. Okayama University, 18 , pp. 117- 134, 1976.
[9] H. Tominaga and A. Yaqub, " Some commutativity properties for rings II," Math. J. Okayama University, :25 , pp. 173- 179, 1983.
[10] H. Tominaga and A. Yaqub, " A commutativity theorem for one sided s-unital rings," Math. J. Okayama University, 25 , pp. 125-128, 1984.
[11] W.Streb, "Zur Struktur Nichtommutativer ringe", Math. J. Okayama University, 31 ), pp. 135- 140, 1989.
[2] H. A. S. Abujabal, M. A. Khan and M. S. Samman, “On commutativity of one-sided s-unital rings," Tamkang J. Math., 23 , pp. 253- 258, 1992.
[3] H. E. Bell, "On two commutativity properties for rings," Mat . .Japonica, 26 , pp. 523- 528, 1981.
[4] I. N. Herstein, " The structure of certain class of rings," Amer. J. Math., 75 , pp. 864- 871, 1953.
[5] Y. Hirano, Y. Kobayashi and H. Tominaga, '' Some polynomial identities and commutativity of s-unital rings," Math. J. Okayama University, 24 , pp. 7- 13, 1982.
[6] Y. Hirano, M. Hongan and H. Tominaga, "Commutativity theorems for certain rings," Math. J. Okayama University, 22 , pp. 65 - 72, 1980.
[7] I. Mogami and M. Hongan, "Note on commutativity of rings,'' Math. J. Okayama University, :20 , pp. 21 - 24, 1978.
[8] H. Tominaga, "On s-unital rings," Math. J. Okayama University, 18 , pp. 117- 134, 1976.
[9] H. Tominaga and A. Yaqub, " Some commutativity properties for rings II," Math. J. Okayama University, :25 , pp. 173- 179, 1983.
[10] H. Tominaga and A. Yaqub, " A commutativity theorem for one sided s-unital rings," Math. J. Okayama University, 25 , pp. 125-128, 1984.
[11] W.Streb, "Zur Struktur Nichtommutativer ringe", Math. J. Okayama University, 31 ), pp. 135- 140, 1989.
Published
2018-04-04
How to Cite
[1]
H. A. S. Abujabal, M. A. Obaid, and M. A. Khan, “On commutativity of rings with constraints involving a nil subset”, Proyecciones (Antofagasta, On line), vol. 15, no. 1, pp. 91-99, Apr. 2018.
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