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

top
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

top

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

top
  1. Abujabal H. A. S., Ashraf M., Some commutativity theorems through a Streb’s classification, Note Mat. 14, No.1 (1994) (to appear). (1994) Zbl0879.16019MR1442008
  2. Ashraf M., On commutativity of one sided s-unital rings with some polynomial constraints, Indian J. Pure and Appl. Math. 25 (1994), 963-967. (1994) Zbl0814.16030MR1294065
  3. Bell H. E., Quadri M. A., Khan M. A., Two commutativity theorems for rings, Rad. Mat. 3 (1994), 255-260. (1994) MR0931981
  4. Bell H. E., Quadri M. A., Ashraf M., Commutativity of rings with some commutator constraints, Rad. Mat. 5 (1989), 223-230. (1989) Zbl0697.16031MR1050891
  5. Chacron M., A commutativity theorem for rings, Proc. Amer. Math. Soc., 59 (1976), 211-216. (1976) Zbl0341.16020MR0414636
  6. Herstein I. N., Two remakrs on commutativity of rings, Canad. J. Math. 7 (1955), 411-412. (1955) MR0071405
  7. Hirano Y., Kobayashi Y., Tominaga H., Some polynomial identities and commutativity of s-unital rings, Math. J. Okayama Univ. 24 (1982), 7-13. (1982) Zbl0487.16023MR0660049
  8. Jacobson N., Structure theory of algebraic algebras of bounded degree, Ann. Math. 46 (1945), 695-707. (1945) MR0014083
  9. Komatsu H., Tominaga H., Chacron’s conditions and commutativity theorems, Math. J. Okayama Univ. 31 (1989), 101-120. (1989) MR1043353
  10. Komatsu H., Tominaga H., Some commutativity theorems for left s-unital rings, Resultate Math. 15 (1989), 335-342. (1989) Zbl0678.16027MR0997069
  11. Komatsu H., Tominaga H., Some commutativity conditions for rings with unity, Resultate Math. 19 (1991), 83-88. (1991) Zbl0776.16017MR1091958
  12. Komatsu H., Nishinaka T., Tominaga H., On commutativity of rings, Rad. Math. 6 (1990), 303-311. (1990) Zbl0718.16031MR1096712
  13. Putcha M. S., Yaqub A., Rings satisfying polynomial constraints, J. Math. Soc., Japan 25 (1973), 115-124. (1973) Zbl0242.16017MR0313312
  14. Quadri M. A., Ashraf M., Khan M. A., A commutativity condition for semiprime ring-II, Bull. Austral. Math. Soc. 33 (1986), 71-73. (1986) MR0823854
  15. Quadri M. A., Ashraf M., Commutativity of generalized Boolean rings, Publ. Math. (Debrecen) 35 (1988), 73-75. (1988) Zbl0657.16020MR0971954
  16. Quadri M. A., Khan M. A., Asma Ali, A commutativity theorem for rings with unity, Soochow J. Math. 15 (1989), 217-227. (1989) MR1045165
  17. Searcoid M. O., MacHale D., Two elementary generalizations for Boolean rings, Amer. Math. Monthly 93 (1986), 121-122. (1986) MR0827587
  18. Streb W., Zur struktur nichtkommutativer Ringe, Math. J. Okayama Univ. 31 (1989), 135-140. (1989) Zbl0702.16022MR1043356
  19. Tominaga H., Yaqub A., Commutativity theorems for rings with constraints involving a commutative subset, Resultate Math. 11 (1987), 186-192. (1987) MR0880201

NotesEmbed ?

top

You must be logged in to post comments.