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

Authors  
Issue Date  1998 
Publisher  World 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. 295310 How to Cite? 
Abstract  A 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 NielsenSchreier property: free nonassociative algebras, free commutative and anticommutative nonassociative algebras, free Lie algebras and superalgebras, and free Lie palgebras and psuperalgebras. 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 Identifier  http://hdl.handle.net/10722/75433 
ISSN  2015 Impact Factor: 0.469 2015 SCImago Journal Rankings: 0.771 
References 
DC Field  Value  Language 

dc.contributor.author  Mikhalev, AA  en_HK 
dc.contributor.author  Yu, JT  en_HK 
dc.date.accessioned  20100906T07:11:04Z   
dc.date.available  20100906T07:11:04Z   
dc.date.issued  1998  en_HK 
dc.identifier.citation  International Journal Of Algebra And Computation, 1998, v. 8 n. 3, p. 295310  en_HK 
dc.identifier.issn  02181967  en_HK 
dc.identifier.uri  http://hdl.handle.net/10722/75433   
dc.description.abstract  A 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 NielsenSchreier property: free nonassociative algebras, free commutative and anticommutative nonassociative algebras, free Lie algebras and superalgebras, and free Lie palgebras and psuperalgebras. 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.language  eng  en_HK 
dc.publisher  World Scientific Publishing Co Pte Ltd. The Journal's web site is located at http://www.worldscinet.com/ijac/ijac.shtml  en_HK 
dc.relation.ispartof  International Journal of Algebra and Computation  en_HK 
dc.title  Test elements, retracts and automorphic orbits of free algebras  en_HK 
dc.type  Article  en_HK 
dc.identifier.openurl  http://library.hku.hk:4550/resserv?sid=HKU:IR&issn=02181967&volume=8&spage=295&epage=310&date=1998&atitle=Test+elements,+retracts+and+automorphic+orbits+of+free+algebras  en_HK 
dc.identifier.email  Yu, JT:yujt@hku.hk  en_HK 
dc.identifier.authority  Yu, JT=rp00834  en_HK 
dc.description.nature  link_to_subscribed_fulltext   
dc.identifier.doi  10.1142/S0218196798000   
dc.identifier.scopus  eid_2s2.00032363775  en_HK 
dc.identifier.hkuros  46810  en_HK 
dc.relation.references  http://www.scopus.com/mlt/select.url?eid=2s2.00032363775&selection=ref&src=s&origin=recordpage  en_HK 
dc.identifier.volume  8  en_HK 
dc.identifier.issue  3  en_HK 
dc.identifier.spage  295  en_HK 
dc.identifier.epage  310  en_HK 
dc.publisher.place  Singapore  en_HK 
dc.identifier.scopusauthorid  Mikhalev, AA=7007020926  en_HK 
dc.identifier.scopusauthorid  Yu, JT=7405530208  en_HK 