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Centralizers in Domains of Gelfand–Kirillov Dimension 2
Author(s) -
Bell Jason P.,
Small Lance W.
Publication year - 2004
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609304003534
Subject(s) - mathematics , algebraically closed field , centralizer and normalizer , pure mathematics , dimension (graph theory) , affine variety , domain (mathematical analysis) , integral domain , field (mathematics) , commutative property , affine transformation , discrete mathematics , mathematical analysis
Given an affine domain of Gelfand–Kirillov dimension 2 over an algebraically closed field, it is shown that the centralizer of any non‐scalar element of this domain is a commutative domain of Gelfand–Kirillov dimension 1 whenever the domain is not polynomial identity. It is shown that the maximal subfields of the quotient division ring of a finitely graded Goldie algebra of Gelfand–Kirillov dimension 2 over a field F all have transcendence degree 1 over F . Finally, centralizers of elements in a finitely graded Goldie domain of Gelfand–Kirillov dimension 2 over an algebraically closed field are considered. In this case, it is shown that the centralizer of a non‐scalar element is an affine commutative domain of Gelfand–Kirillov dimension 1. 2000 Mathematics Subject Classification 16P90.