| Title:
|
Commutativity of rings through a Streb’s result (English) |
| Author:
|
Khan, Moharram A. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
50 |
| Issue:
|
4 |
| Year:
|
2000 |
| Pages:
|
791-801 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we investigate commutativity of rings with unity satisfying any one of the properties: \[ \begin{aligned} &\lbrace 1- g(yx^{m}) \rbrace \ [yx^{m} - x^{r} f (yx^{m}) \ x^s, x] \lbrace 1- h (yx^{m}) \rbrace = 0, \\&\lbrace 1- g(yx^{m}) \rbrace \ [x^{m} y - x^{r} f (yx^{m}) x^{s}, x] \lbrace 1- h (yx^{m}) \rbrace = 0, \\&y^{t} [x,y^{n}] = g (x) [f (x), y] h (x)\ {\mathrm and} \ \ [x,y^{n}] \ y^{t} = g (x) [f (x), y] h (x) \end{aligned} \] for some $f(X)$ in $X^{2} {\mathbb Z}[X]$ and $g(X)$, $ h(X)$ in ${\mathbb Z} [X]$, where $m \ge 0$, $ r \ge 0$, $ s \ge 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 has been studied. These results generalize a number of commutativity theorems established recently. (English) |
| Keyword:
|
commutators |
| Keyword:
|
division rings |
| Keyword:
|
factorsubrings |
| Keyword:
|
polynomial identities |
| Keyword:
|
torsion-free rings |
| MSC:
|
16R50 |
| MSC:
|
16U70 |
| MSC:
|
16U80 |
| idZBL:
|
Zbl 1079.16504 |
| idMR:
|
MR1792970 |
| . |
| Date available:
|
2009-09-24T10:37:53Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127610 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
[9] W. Streb: Zur Struktur nichtkommutativer Ringe.Math. J. Okayama Univ. 31 (1989), 135–140. Zbl 0702.16022, MR 1043356 |
| Reference:
|
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| Reference:
|
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| . |
