| Title:
|
A commutativity theorem for associative rings (English) |
| Author:
|
Ashraf, Mohammad |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
31 |
| Issue:
|
3 |
| Year:
|
1995 |
| Pages:
|
201-204 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $m > 1, s\geq 1$ be fixed positive integers, and let $R$ be a ring with unity $1$ in which for every $x$ in $R$ there exist integers $p = p(x) \geq 0, q = q(x) \geq 0, n = n(x) \geq 0, r = r(x) \geq 0 $ such that either $ x^{p}[x^{n},y]x^{q} = x^{r}[x,y^{m}]y^{s} $ or $ x^{p}[x^{n},y]x^{q} = y^{s}[x,y^{m}]x^{r} $ for all $ y \in R $. In the present paper it is shown that $R$ is commutative if it satisfies the property $Q(m)$ (i.e. for all $x,y \in R, m[x,y] = 0$ implies $[x,y] = 0$). (English) |
| Keyword:
|
polynomial identity |
| Keyword:
|
nilpotent element |
| Keyword:
|
commutator ideal |
| Keyword:
|
associative ring |
| Keyword:
|
torsion free ring |
| Keyword:
|
center |
| Keyword:
|
commutativity |
| MSC:
|
16R50 |
| MSC:
|
16U70 |
| MSC:
|
16U80 |
| idZBL:
|
Zbl 0839.16030 |
| idMR:
|
MR1368258 |
| . |
| Date available:
|
2008-06-06T21:28:55Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107540 |
| . |
| Reference:
|
[1] Abujabal H. A. S.: On commutativity of left s-unital rings.Acta Sci. Math. (Szeged) 56 (1992), 51-62. Zbl 0806.16034, MR 1204738 |
| Reference:
|
[2] Abujabal H. A. S., Obaid M. A.: Some commutativity theorems for right s-unital rings.Math. Japonica, 37, No. 3 (1992), 591-600. Zbl 0767.16010, MR 1162474 |
| Reference:
|
[3] Ashraf M., Quadri M. A.: On commutativity of associative rings with constraints involving a subset.Rad. Mat.5 (1989), 141-149. Zbl 0683.16025, MR 1012730 |
| Reference:
|
[4] Ashraf M., Jacob V. W.: On certain polynomial identities implying commutativity for rings.(submitted). Zbl 0988.16518 |
| Reference:
|
[5] Bell H. E.: On the power map and ring commutativity.Canad. Math. Bull. 21 (1978), 399-404. Zbl 0403.16024, MR 0523579 |
| Reference:
|
[6] Bell H. E.: Commutativity of rings with constraints on commutators.Resultate der Math. 8 (1985), 123-131. Zbl 0606.16023, MR 0828934 |
| Reference:
|
[7] Hermanci A.: Two elementary commutativity theorems for rings.Acta Math. Acad.Sci. Hungar. 29 (1977),23-29. MR 0444712 |
| Reference:
|
[8] Jacobson N.: Structure of rings.37 (Amer. Math. Soc. Colloq. Publ. Providence, 1956). Zbl 0073.02002, MR 0081264 |
| Reference:
|
[9] Kezlan T. P.: A note on commutativity of semi-prime PI- rings.Math. Japonica 27 (1982) 267-268. MR 0655230 |
| Reference:
|
[10] Kezlan T. P.: A commutativity theorem involving certain polynomial constraints.Math. Japonica 36, No. 4 (1991),785-789. Zbl 0735.16021, MR 1120461 |
| Reference:
|
[11] Kezlan T. P.: On commutativity theorems for PI-rings with unity.Tamkang J. math. 24 No. 1 (1993), 29-36. MR 1215242 |
| Reference:
|
[12] Komatsu H.: A commutativity theorem for rings.Math. J. Okayama Univ. 26 (1984), 135-139. Zbl 0568.16017, MR 0779780 |
| Reference:
|
[13] Komatsu H.: A commutativity theorem for rings-II.Osaka J. Math. 22 (1985), 811-814. Zbl 0575.16017, MR 0815449 |
| Reference:
|
[14] Nicholson W. K., Yaqub A.: A commutativity theorem for rings and groups.Canad. Math. Bull. 22 (1979), 419-423. Zbl 0605.16020, MR 0563755 |
| Reference:
|
[15] Psomopoulos E.: A commutativity theorem for rings involving a subset of the ring.Glasnik Mat. 18 (1983), 231-236. Zbl 0528.16017, MR 0733162 |
| Reference:
|
[16] Psomopoulos E.: Commutativity theorems for rings and groups with constraints on commutators.Internat. J. Math. & Math. Sci. 7 No. 3(1984), 513-517. Zbl 0561.16013, MR 0771600 |
| . |
