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Article: Factor algebras of free algebras: On a problem of G. Bergman

TitleFactor algebras of free algebras: On a problem of G. Bergman
Authors
Issue Date2003
Citation
Bulletin Of The London Mathematical Society, 2003, v. 35 n. 5, p. 706-710 How to Cite?
AbstractLet An = K〈X1,...,Xn〉 be a free associative algebra over a field K. In this paper, examples are given of elements u ∈ An, n ≥ 3, such that the factor algebra of An over the ideal generated by u is isomorphic to An-1, and yet u is not a primitive element of An (that is, it cannot be taken to x1 by an automorphism of An). 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.
Persistent Identifierhttp://hdl.handle.net/10722/156115
ISSN
2015 Impact Factor: 0.789
2015 SCImago Journal Rankings: 1.474
ISI Accession Number ID
References

 

DC FieldValueLanguage
dc.contributor.authorShpilrain, Ven_US
dc.contributor.authorYu, JTen_US
dc.date.accessioned2012-08-08T08:40:27Z-
dc.date.available2012-08-08T08:40:27Z-
dc.date.issued2003en_US
dc.identifier.citationBulletin Of The London Mathematical Society, 2003, v. 35 n. 5, p. 706-710en_US
dc.identifier.issn0024-6093en_US
dc.identifier.urihttp://hdl.handle.net/10722/156115-
dc.description.abstractLet An = K〈X1,...,Xn〉 be a free associative algebra over a field K. In this paper, examples are given of elements u ∈ An, n ≥ 3, such that the factor algebra of An over the ideal generated by u is isomorphic to An-1, and yet u is not a primitive element of An (that is, it cannot be taken to x1 by an automorphism of An). 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.en_US
dc.languageengen_US
dc.relation.ispartofBulletin of the London Mathematical Societyen_US
dc.titleFactor algebras of free algebras: On a problem of G. Bergmanen_US
dc.typeArticleen_US
dc.identifier.emailYu, JT:yujt@hku.hken_US
dc.identifier.authorityYu, JT=rp00834en_US
dc.description.naturelink_to_subscribed_fulltexten_US
dc.identifier.doi10.1112/S002460930300225Xen_US
dc.identifier.scopuseid_2-s2.0-0042916444en_US
dc.relation.referenceshttp://www.scopus.com/mlt/select.url?eid=2-s2.0-0042916444&selection=ref&src=s&origin=recordpageen_US
dc.identifier.volume35en_US
dc.identifier.issue5en_US
dc.identifier.spage706en_US
dc.identifier.epage710en_US
dc.identifier.isiWOS:000185267200015-
dc.publisher.placeUnited Kingdomen_US
dc.identifier.scopusauthoridShpilrain, V=6603904879en_US
dc.identifier.scopusauthoridYu, JT=7405530208en_US

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