Commutativity of rings through a Streb’s result

Khan, Moharram A.. "Commutativity of rings through a Streb’s result." Czechoslovak Mathematical Journal 50.4 (2000): 791-801. <http://eudml.org/doc/30600>.

@article{Khan2000,
abstract = {In this paper we investigate commutativity of rings with unity satisfying any one of the properties: \[ \begin\{aligned\} &\lbrace 1- g(yx^\{m\}) \rbrace \ [yx^\{m\} - x^\{r\} f (yx^\{m\}) \ x^s, x] \lbrace 1- h (yx^\{m\}) \rbrace = 0, \\&\lbrace 1- g(yx^\{m\}) \rbrace \ [x^\{m\} y - x^\{r\} f (yx^\{m\}) x^\{s\}, x] \lbrace 1- h (yx^\{m\}) \rbrace = 0, \\&y^\{t\} [x,y^\{n\}] = g (x) [f (x), y] h (x)\ \{\mathrm \{a\}nd\} \ \ [x,y^\{n\}] \ y^\{t\} = g (x) [f (x), y] h (x) \end\{aligned\} \] 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.},
author = {Khan, Moharram A.},
journal = {Czechoslovak Mathematical Journal},
keywords = {commutators; division rings; factorsubrings; polynomial identities; torsion-free rings; commutator constraints; division rings; factor subrings; polynomial identities; torsion-free rings; commutativity theorems},
language = {eng},
number = {4},
pages = {791-801},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Commutativity of rings through a Streb’s result},
url = {http://eudml.org/doc/30600},
volume = {50},
year = {2000},
}

TY - JOUR
AU - Khan, Moharram A.
TI - Commutativity of rings through a Streb’s result
JO - Czechoslovak Mathematical Journal
PY - 2000
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 50
IS - 4
SP - 791
EP - 801
AB - In this paper we investigate commutativity of rings with unity satisfying any one of the properties: \[ \begin{aligned} &\lbrace 1- g(yx^{m}) \rbrace \ [yx^{m} - x^{r} f (yx^{m}) \ x^s, x] \lbrace 1- h (yx^{m}) \rbrace = 0, \\&\lbrace 1- g(yx^{m}) \rbrace \ [x^{m} y - x^{r} f (yx^{m}) x^{s}, x] \lbrace 1- h (yx^{m}) \rbrace = 0, \\&y^{t} [x,y^{n}] = g (x) [f (x), y] h (x)\ {\mathrm {a}nd} \ \ [x,y^{n}] \ y^{t} = g (x) [f (x), y] h (x) \end{aligned} \] 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.
LA - eng
KW - commutators; division rings; factorsubrings; polynomial identities; torsion-free rings; commutator constraints; division rings; factor subrings; polynomial identities; torsion-free rings; commutativity theorems
UR - http://eudml.org/doc/30600
ER -