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Article: Test elements, retracts and automorphic orbits of free algebras

TitleTest elements, retracts and automorphic orbits of free algebras
Authors
Issue Date1998
PublisherWorld Scientific Publishing Co Pte Ltd. The Journal's web site is located at http://www.worldscinet.com/ijac/ijac.shtml
Citation
International Journal Of Algebra And Computation, 1998, v. 8 n. 3, p. 295-310 How to Cite?
AbstractA nonzero element a of an algebra A is called a test element if for any endomorphism φ of A it follows from φ(a) = a that φ is an automorphism of the algebra A. A subalgebra B of A is a retract if there is an ideal I of A such that A = B ⊕ I. We consider the main types of free algebras with the Nielsen-Schreier property: free nonassociative algebras, free commutative and anti-commutative nonassociative algebras, free Lie algebras and superalgebras, and free Lie p-algebras and p-superalgebras. For any free algebra F of finite rank of such type we prove that an element u is a test element if and only if it does not belong to any proper retract of F. Test elements for monomorphisms of F are exactly elements that are not contained in proper free factors of F. These results give analogs of Turner's results on test elements of free groups. We also characterize retracts of the algebra F. We prove that if some endomorphism φ preserve the automorphic orbit of some nonzero element of F, then φ is a monomorphism. For free Lie algebras and superalgebras over a field of characteristic zero and for free Lie p-(super)algebras over a field of prime characteristic p we show that in this situation φ is an automorphism. We discuss some related topics and formulate open problems.
Persistent Identifierhttp://hdl.handle.net/10722/75433
ISSN
2015 Impact Factor: 0.469
2015 SCImago Journal Rankings: 0.771
References

 

DC FieldValueLanguage
dc.contributor.authorMikhalev, AAen_HK
dc.contributor.authorYu, JTen_HK
dc.date.accessioned2010-09-06T07:11:04Z-
dc.date.available2010-09-06T07:11:04Z-
dc.date.issued1998en_HK
dc.identifier.citationInternational Journal Of Algebra And Computation, 1998, v. 8 n. 3, p. 295-310en_HK
dc.identifier.issn0218-1967en_HK
dc.identifier.urihttp://hdl.handle.net/10722/75433-
dc.description.abstractA nonzero element a of an algebra A is called a test element if for any endomorphism φ of A it follows from φ(a) = a that φ is an automorphism of the algebra A. A subalgebra B of A is a retract if there is an ideal I of A such that A = B ⊕ I. We consider the main types of free algebras with the Nielsen-Schreier property: free nonassociative algebras, free commutative and anti-commutative nonassociative algebras, free Lie algebras and superalgebras, and free Lie p-algebras and p-superalgebras. For any free algebra F of finite rank of such type we prove that an element u is a test element if and only if it does not belong to any proper retract of F. Test elements for monomorphisms of F are exactly elements that are not contained in proper free factors of F. These results give analogs of Turner's results on test elements of free groups. We also characterize retracts of the algebra F. We prove that if some endomorphism φ preserve the automorphic orbit of some nonzero element of F, then φ is a monomorphism. For free Lie algebras and superalgebras over a field of characteristic zero and for free Lie p-(super)algebras over a field of prime characteristic p we show that in this situation φ is an automorphism. We discuss some related topics and formulate open problems.en_HK
dc.languageengen_HK
dc.publisherWorld Scientific Publishing Co Pte Ltd. The Journal's web site is located at http://www.worldscinet.com/ijac/ijac.shtmlen_HK
dc.relation.ispartofInternational Journal of Algebra and Computationen_HK
dc.titleTest elements, retracts and automorphic orbits of free algebrasen_HK
dc.typeArticleen_HK
dc.identifier.openurlhttp://library.hku.hk:4550/resserv?sid=HKU:IR&issn=0218-1967&volume=8&spage=295&epage=310&date=1998&atitle=Test+elements,+retracts+and+automorphic+orbits+of+free+algebrasen_HK
dc.identifier.emailYu, JT:yujt@hku.hken_HK
dc.identifier.authorityYu, JT=rp00834en_HK
dc.description.naturelink_to_subscribed_fulltext-
dc.identifier.doi10.1142/S0218196798000-
dc.identifier.scopuseid_2-s2.0-0032363775en_HK
dc.identifier.hkuros46810en_HK
dc.relation.referenceshttp://www.scopus.com/mlt/select.url?eid=2-s2.0-0032363775&selection=ref&src=s&origin=recordpageen_HK
dc.identifier.volume8en_HK
dc.identifier.issue3en_HK
dc.identifier.spage295en_HK
dc.identifier.epage310en_HK
dc.publisher.placeSingaporeen_HK
dc.identifier.scopusauthoridMikhalev, AA=7007020926en_HK
dc.identifier.scopusauthoridYu, JT=7405530208en_HK

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