# A commutativity theorem for associative rings

Archivum Mathematicum (1995)

• Volume: 031, Issue: 3, page 201-204
• ISSN: 0044-8753

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## Abstract

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Let $m>1,s\ge 1$ be fixed positive integers, and let $R$ be a ring with unity $1$ in which for every $x$ in $R$ there exist integers $p=p\left(x\right)\ge 0,q=q\left(x\right)\ge 0,n=n\left(x\right)\ge 0,r=r\left(x\right)\ge 0$ such that either ${x}^{p}\left[{x}^{n},y\right]{x}^{q}={x}^{r}\left[x,{y}^{m}\right]{y}^{s}$ or ${x}^{p}\left[{x}^{n},y\right]{x}^{q}={y}^{s}\left[x,{y}^{m}\right]{x}^{r}$ for all $y\in R$. In the present paper it is shown that $R$ is commutative if it satisfies the property $Q\left(m\right)$ (i.e. for all $x,y\in R,m\left[x,y\right]=0$ implies $\left[x,y\right]=0$).

## How to cite

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Ashraf, Mohammad. "A commutativity theorem for associative rings." Archivum Mathematicum 031.3 (1995): 201-204. <http://eudml.org/doc/247699>.

@article{Ashraf1995,
abstract = {Let $m > 1, s\ge 1$ be fixed positive integers, and let $R$ be a ring with unity $1$ in which for every $x$ in $R$ there exist integers $p = p(x) \ge 0, q = q(x) \ge 0, n = n(x) \ge 0, r = r(x) \ge 0$ 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 $[x,y] = 0$).},
journal = {Archivum Mathematicum},
keywords = {polynomial identity; nilpotent element; commutator ideal; associative ring; torsion free ring; center; commutativity; commutativity theorem; commutator constraints},
language = {eng},
number = {3},
pages = {201-204},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {A commutativity theorem for associative rings},
url = {http://eudml.org/doc/247699},
volume = {031},
year = {1995},
}

TY - JOUR
TI - A commutativity theorem for associative rings
JO - Archivum Mathematicum
PY - 1995
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 031
IS - 3
SP - 201
EP - 204
AB - Let $m > 1, s\ge 1$ be fixed positive integers, and let $R$ be a ring with unity $1$ in which for every $x$ in $R$ there exist integers $p = p(x) \ge 0, q = q(x) \ge 0, n = n(x) \ge 0, r = r(x) \ge 0$ 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 $[x,y] = 0$).
LA - eng
KW - polynomial identity; nilpotent element; commutator ideal; associative ring; torsion free ring; center; commutativity; commutativity theorem; commutator constraints
UR - http://eudml.org/doc/247699
ER -

## References

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11. Kezlan T. P., On commutativity theorems for PI-rings with unity, Tamkang J. math. 24 No. 1 (1993), 29-36. (1993) MR1215242
12. Komatsu H., A commutativity theorem for rings, Math. J. Okayama Univ. 26 (1984), 135-139. (1984) Zbl0568.16017MR0779780
13. Komatsu H., A commutativity theorem for rings-II, Osaka J. Math. 22 (1985), 811-814. (1985) Zbl0575.16017MR0815449
14. Nicholson W. K., Yaqub A., A commutativity theorem for rings and groups, Canad. Math. Bull. 22 (1979), 419-423. (1979) Zbl0605.16020MR0563755
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