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Factor Algebras of Free Algebras: On a Problem of G. Bergman
Author(s) -
Shpilrain Vladimir,
Yu JieTai
Publication year - 2003
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/s002460930300225x
Subject(s) - mathematics , automorphism , ideal (ethics) , conjecture , embedding , field (mathematics) , pure mathematics , mathematics subject classification , jordan algebra , algebraic variety , affine transformation , polynomial , algebra over a field , algebraic number , combinatorics , algebra representation , computer science , mathematical analysis , philosophy , epistemology , artificial intelligence
Let A n = K 〈 x 1 ,…, x n 〉 be a free associative algebra over a field K . In this paper, examples are given of elements u ∈ A n , n ⩾ 3, such that the factor algebra of A n over the ideal generated by u is isomorphic to A n −1 , and yet u is not a primitive element of A n (that is, it cannot be taken to x 1 by an automorphism of A n ). If the characteristic of the ground field K is 0, this yields a negative answer to a question of G. Bergman. On the other hand, by a result of Drensky and Yu, there is no such example for n = 2. It should be noted that a similar question for polynomial algebras, known as the embedding conjecture of Abhyankar and Sathaye, is a major open problem in affine algebraic geometry. 2000 Mathematics Subject Classification 16S10, 16W20 (primary); 14A05, 13B25 (secondary).