Centralizers on prime and semiprime rings

Joso Vukman

Commentationes Mathematicae Universitatis Carolinae (1997)

  • Volume: 38, Issue: 2, page 231-240
  • ISSN: 0010-2628

Abstract

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The purpose of this paper is to investigate identities satisfied by centralizers on prime and semiprime rings. We prove the following result: Let R be a noncommutative prime ring of characteristic different from two and let S and T be left centralizers on R . Suppose that [ S ( x ) , T ( x ) ] S ( x ) + S ( x ) [ S ( x ) , T ( x ) ] = 0 is fulfilled for all x ∈ R . If S ≠ 0 ( T ≠ 0 ) then there exists λ from the extended centroid of R such that T = λ S ( S = λ T ) .

How to cite

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Vukman, Joso. "Centralizers on prime and semiprime rings." Commentationes Mathematicae Universitatis Carolinae 38.2 (1997): 231-240. <http://eudml.org/doc/248087>.

@article{Vukman1997,
abstract = {The purpose of this paper is to investigate identities satisfied by centralizers on prime and semiprime rings. We prove the following result: Let $R$ be a noncommutative prime ring of characteristic different from two and let $S$ and $T$ be left centralizers on $R$. Suppose that $[S(x),T(x)]S(x)+S(x)[S(x),T(x)]=0$ is fulfilled for all $x\in R$. If $S\ne 0$$(T\ne 0)$ then there exists $\lambda $ from the extended centroid of $R$ such that $T=\lambda S$$(S=\lambda T)$.},
author = {Vukman, Joso},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {prime ring; semiprime ring; extended centroid; derivation; Jordan derivation; left (right) centralizer; Jordan left (right) centralizer; commuting mapping; centralizing mapping; prime rings; semiprime rings; extended centroids; derivations; Jordan derivations; left centralizers; Jordan right centralizers; commuting mappings; centralizing mappings},
language = {eng},
number = {2},
pages = {231-240},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Centralizers on prime and semiprime rings},
url = {http://eudml.org/doc/248087},
volume = {38},
year = {1997},
}

TY - JOUR
AU - Vukman, Joso
TI - Centralizers on prime and semiprime rings
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1997
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 38
IS - 2
SP - 231
EP - 240
AB - The purpose of this paper is to investigate identities satisfied by centralizers on prime and semiprime rings. We prove the following result: Let $R$ be a noncommutative prime ring of characteristic different from two and let $S$ and $T$ be left centralizers on $R$. Suppose that $[S(x),T(x)]S(x)+S(x)[S(x),T(x)]=0$ is fulfilled for all $x\in R$. If $S\ne 0$$(T\ne 0)$ then there exists $\lambda $ from the extended centroid of $R$ such that $T=\lambda S$$(S=\lambda T)$.
LA - eng
KW - prime ring; semiprime ring; extended centroid; derivation; Jordan derivation; left (right) centralizer; Jordan left (right) centralizer; commuting mapping; centralizing mapping; prime rings; semiprime rings; extended centroids; derivations; Jordan derivations; left centralizers; Jordan right centralizers; commuting mappings; centralizing mappings
UR - http://eudml.org/doc/248087
ER -

References

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  7. Herstein I.N., Jordan derivations on prime rings, Proc. Amer. Math. Soc. 8 (1957), 1104-1110. (1957) MR0095864
  8. Herstein I.N., Rings with involution, Univ. of Chicago Press, Chicago, 1976. Zbl0495.16007MR0442017
  9. Martindale W.S., Prime rings satisfying a generalized polynomial identity, Journal of Algebra 12 (1969), 576-584. (1969) Zbl0175.03102MR0238897
  10. Posner E., Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093-1100. (1957) MR0095863
  11. Zalar B., On centralizers of semiprime rings, Comment. Math. Univ. Carolinae 32 (1991), 609-614. (1991) Zbl0746.16011MR1159807

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