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Centralizers and Multilinear Polynomials in Non‐Commutative Rings
Author(s) -
Felzenszwalb Bernardo,
Giambruno Antonino
Publication year - 1979
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-19.3.417
Subject(s) - degree (music) , mathematics , center (category theory) , combinatorics , multilinear map , commutative ring , polynomial ring , polynomial , commutative property , prime (order theory) , ring (chemistry) , discrete mathematics , pure mathematics , physics , crystallography , chemistry , mathematical analysis , organic chemistry , acoustics
Let C be a commutative ring with 1, R a C ‐algebra, and f ( x 1 , …, x d ) a multilinear polynomial in C { x 1 , …, x d } such that for some coefficient α of f , α R ≠ 0. Let T = {α ∈ R \ af n ( r 1 , …, r d ) = f n ( r 1 , …, r d ) a , all r 1 , …, r d ∈ R , for some n = n ( a , r 1 , …, r d ) ⩾ 1} THEOREM. Let R be a prime ring with no non‐zero nil right ideals. If char R = p ≠ 0, assume that f is not an identity in p × p matrices in characteristic p. Then either T is the center of R, or R is PI, PI‐degree ( R ) ⩽ 1/2[deg( f )+2], and the values of f are power central . In case f = S d , the standard polynomial of degree d, if T is not the center of R, then d = 1 and (i) if d = 3 and char R ≠ 3 or d = 3 then S d vanishes in R ; (ii) if d = 3 and char R = 3 or d = 2 then S d 2 is central in R and PI‐degree (R) = 2