Commutativity of rings with polynomial constraints

Moharram A. Khan

Czechoslovak Mathematical Journal (2002)

  • Volume: 52, Issue: 2, page 401-413
  • ISSN: 0011-4642

Abstract

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Let p , q and r be fixed non-negative integers. In this note, it is shown that if R is left (right) s -unital ring satisfying [ f ( x p y q ) - x r y , x ] = 0 ( [ f ( x p y q ) - y x r , x ] = 0 , respectively) where f ( λ ) ∈ λ 2 ℤ [ λ ] , then R is commutative. Moreover, commutativity of R is also obtained under different sets of constraints on integral exponents. Also, we provide some counterexamples which show that the hypotheses are not altogether superfluous. Thus, many well-known commutativity theorems become corollaries of our results.

How to cite

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Khan, Moharram A.. "Commutativity of rings with polynomial constraints." Czechoslovak Mathematical Journal 52.2 (2002): 401-413. <http://eudml.org/doc/30710>.

@article{Khan2002,
abstract = {Let $p$, $ q$ and $r$ be fixed non-negative integers. In this note, it is shown that if $R$ is left (right) $s$-unital ring satisfying $[f(x^py^q) - x^ry, x] = 0$ ($[f(x^py^q) - yx^r, x] = 0$, respectively) where $f(\lambda ) \in \{\lambda \}^2\{\mathbb \{Z\}\}[\lambda ]$, then $R$ is commutative. Moreover, commutativity of $R$ is also obtained under different sets of constraints on integral exponents. Also, we provide some counterexamples which show that the hypotheses are not altogether superfluous. Thus, many well-known commutativity theorems become corollaries of our results.},
author = {Khan, Moharram A.},
journal = {Czechoslovak Mathematical Journal},
keywords = {automorphism; commutativity; local ring; polynomial identity; $s$-unital ring; automorphisms; commutativity theorems; local rings; polynomial identities; -unital rings; polynomial constraints},
language = {eng},
number = {2},
pages = {401-413},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Commutativity of rings with polynomial constraints},
url = {http://eudml.org/doc/30710},
volume = {52},
year = {2002},
}

TY - JOUR
AU - Khan, Moharram A.
TI - Commutativity of rings with polynomial constraints
JO - Czechoslovak Mathematical Journal
PY - 2002
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 52
IS - 2
SP - 401
EP - 413
AB - Let $p$, $ q$ and $r$ be fixed non-negative integers. In this note, it is shown that if $R$ is left (right) $s$-unital ring satisfying $[f(x^py^q) - x^ry, x] = 0$ ($[f(x^py^q) - yx^r, x] = 0$, respectively) where $f(\lambda ) \in {\lambda }^2{\mathbb {Z}}[\lambda ]$, then $R$ is commutative. Moreover, commutativity of $R$ is also obtained under different sets of constraints on integral exponents. Also, we provide some counterexamples which show that the hypotheses are not altogether superfluous. Thus, many well-known commutativity theorems become corollaries of our results.
LA - eng
KW - automorphism; commutativity; local ring; polynomial identity; $s$-unital ring; automorphisms; commutativity theorems; local rings; polynomial identities; -unital rings; polynomial constraints
UR - http://eudml.org/doc/30710
ER -

References

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  1. 10.1023/A:1022197800695, Georgian Math.  J. 5 (1998), 301–314. (1998) MR1639061DOI10.1023/A:1022197800695
  2. A commutativity theorem for one sided s -unital rings, Pure Math. Appl. 1 (1990), 109–116. (1990) MR1095008
  3. Two commutativity theorems for rings, Rad. Mat. 3 (1987), 255–260. (1987) MR0931981
  4. 10.4153/CMB-1978-070-x, Canad. Math. Bull. 21 (1978), 399–404. (1978) Zbl0403.16024MR0523579DOI10.4153/CMB-1978-070-x
  5. 10.1007/BF01228168, Arch. Math. 24 (1973), 34-38. (1973) Zbl0251.16021MR0320090DOI10.1007/BF01228168
  6. 10.1307/mmj/1028998511, Michigan Math.  J. 8 (1961), 29–32. (1961) Zbl0096.25701MR0118741DOI10.1307/mmj/1028998511
  7. 10.4153/CJM-1955-044-2, Canad. J.  Math. 7 (1955), 411–412. (1955) MR0071405DOI10.4153/CJM-1955-044-2
  8. Some polynomial identities and commutativity of s -unital rings, Math. J.  Okayama Univ. 24 (1982), 7–13. (1982) MR0660049
  9. Structure of Rings, Amer. Math. Soc. Colloq. Publ., Providence, 1956. (1956) Zbl0073.02002MR0081264
  10. A note on commutativity of semiprime PI-rings, Math. Japon. 27 (1982), 267–268. (1982) Zbl0481.16013MR0655230
  11. 10.1023/A:1022464612374, Czecholoslovak Math. J. 50 (2000), 791–801. (2000) Zbl1079.16504MR1792970DOI10.1023/A:1022464612374
  12. On commutativity of rings, Rad. Mat. 6 (1990), 303–311. (1990) MR1096712
  13. Chacron’s condition and commutativity theorems, Math. J. Okayama Univ. 31 (1989), 101–120. (1989) MR1043353
  14. 10.1007/BF03322621, Resultate der Math. 15 (1989), 335–342. (1989) Zbl0678.16027MR0997069DOI10.1007/BF03322621
  15. 10.4153/CMB-1979-055-9, Canad. Math. Bull. 22 (1979), 419–423. (1979) MR0563755DOI10.4153/CMB-1979-055-9
  16. A commutativity theorem for rings, Math. Japon. 29 (1984), 371–373. (1984) Zbl0548.16029MR0752233
  17. Rings satisfying polynomial constraints, J. Math. Soc. Japan 25 (1973), 115–124. (1973) MR0313312
  18. Zur Struktur nichtkommutativer Ringe, Math. J.  Okayama Univ. 31 (1989), 135–140. (1989) Zbl0702.16022MR1043356
  19. Über einen Satz von Herstein und Nakayama, Rend. Sem. Mat. Univ. Podova 64 (1981), 151–171. (1981) Zbl0474.16024MR0636633

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