Commutativity of associative rings through a Streb's classification

Mohammad Ashraf

Archivum Mathematicum (1997)

  • Volume: 033, Issue: 4, page 315-321
  • ISSN: 0044-8753

Abstract

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Let m 0 , r 0 , s 0 , q 0 be fixed integers. Suppose that R is an associative ring with unity 1 in which for each x , y R there exist polynomials f ( X ) X 2 Z Z [ X ] , g ( X ) , h ( X ) X Z Z [ X ] such that { 1 - g ( y x m ) } [ x , x r y - x s f ( y x m ) x q ] { 1 - h ( y x m ) } = 0 . Then R is commutative. Further, result is extended to the case when the integral exponents in the above property depend on the choice of x and y . Finally, commutativity of one sided s-unital ring is also obtained when R satisfies some related ring properties.

How to cite

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Ashraf, Mohammad. "Commutativity of associative rings through a Streb's classification." Archivum Mathematicum 033.4 (1997): 315-321. <http://eudml.org/doc/248034>.

@article{Ashraf1997,
abstract = {Let $m \ge 0, ~r \ge 0, ~s \ge 0, ~q \ge 0$ be fixed integers. Suppose that $R$ is an associative ring with unity $1$ in which for each $x,y \in R$ there exist polynomials $f(X) \in X^\{2\} \mbox\{$Z \hspace\{-6.25958pt\} Z$\}[X], ~g(X), ~h(X) \in X \mbox\{$Z \hspace\{-6.25958pt\} Z$\}[X]$ such that $\lbrace 1-g (yx^\{m\}) \rbrace [x, ~x^\{r\}y ~-~ x^\{s\}f (y x^\{m\}) x^\{q\}] \lbrace 1-h(yx^\{m\}) \rbrace ~=~ 0$. Then $R$ is commutative. Further, result is extended to the case when the integral exponents in the above property depend on the choice of $x$ and $y$. Finally, commutativity of one sided s-unital ring is also obtained when $R$ satisfies some related ring properties.},
author = {Ashraf, Mohammad},
journal = {Archivum Mathematicum},
keywords = {factorsubring; s-unital ring; commutativity; commutator; associative ring; -unital rings; commutativity theorems; commutators},
language = {eng},
number = {4},
pages = {315-321},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Commutativity of associative rings through a Streb's classification},
url = {http://eudml.org/doc/248034},
volume = {033},
year = {1997},
}

TY - JOUR
AU - Ashraf, Mohammad
TI - Commutativity of associative rings through a Streb's classification
JO - Archivum Mathematicum
PY - 1997
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 033
IS - 4
SP - 315
EP - 321
AB - Let $m \ge 0, ~r \ge 0, ~s \ge 0, ~q \ge 0$ be fixed integers. Suppose that $R$ is an associative ring with unity $1$ in which for each $x,y \in R$ there exist polynomials $f(X) \in X^{2} \mbox{$Z \hspace{-6.25958pt} Z$}[X], ~g(X), ~h(X) \in X \mbox{$Z \hspace{-6.25958pt} Z$}[X]$ such that $\lbrace 1-g (yx^{m}) \rbrace [x, ~x^{r}y ~-~ x^{s}f (y x^{m}) x^{q}] \lbrace 1-h(yx^{m}) \rbrace ~=~ 0$. Then $R$ is commutative. Further, result is extended to the case when the integral exponents in the above property depend on the choice of $x$ and $y$. Finally, commutativity of one sided s-unital ring is also obtained when $R$ satisfies some related ring properties.
LA - eng
KW - factorsubring; s-unital ring; commutativity; commutator; associative ring; -unital rings; commutativity theorems; commutators
UR - http://eudml.org/doc/248034
ER -

References

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