Finitely determined members of varieties of groups and rings

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dc.contributor.authorBelegradek, O
dc.date.accessioned2024-07-18T20:40:03Z
dc.date.available2024-07-18T20:40:03Z
dc.date.issued2000
dc.departmentİstanbul Bilgi Üniversitesien_US
dc.description.abstractA finitely generated algebra A in a variety V is called finitely determined in V if there exists a finite V-consistent set of equalities and inequalities in an alphabet containing the generating set of A, which, together with the identities of V, yields all relations and non-relations of A. Obviously, if the equational theory of V is recursively enumerable then any finitely determined algebra in V has solvable word problem. The known algebraic characterizations of groups and semigroups with solvable word problem imply that in the varieties of all groups and all semigroups the members with solvable word problem are finitely determined. We construct a finitely generated center-by-metabelian group with solvable word problem, which is not finitely determined in every group variety V with ZU(2) subset of or equal to V subset of or equal to U-3. We show that every extension of a finitely generated abelian group by a finite group from a variety W is finitely determined in every variety V superset of or equal to ZU(2)W. However, in any abelian-by-nilpotent variety no infinite group is finitely determined; moreover, in every variety, in which all finitely presented algebras are residually finite, each finitely determined algebra is finite. In the variety of all associative linear algebras over a finitely generated field every member with solvable word problem is finitely determined. We construct an example, which shows that for the variety of all associative rings it is not true; however, in this variety each torsion-free member with solvable word problem is finitely determined. (C) 2000 Academic Press.en_US
dc.identifier.doi10.1006/jabr.2000.8287
dc.identifier.endpage602en_US
dc.identifier.issn0021-8693
dc.identifier.issue2en_US
dc.identifier.scopus2-s2.0-0034658783en_US
dc.identifier.scopusqualityQ1en_US
dc.identifier.startpage586en_US
dc.identifier.urihttps://doi.org/10.1006/jabr.2000.8287
dc.identifier.urihttps://hdl.handle.net/11411/6952
dc.identifier.volume228en_US
dc.identifier.wosWOS:000087678800013en_US
dc.identifier.wosqualityQ2en_US
dc.indekslendigikaynakWeb of Scienceen_US
dc.indekslendigikaynakScopusen_US
dc.language.isoenen_US
dc.publisherAcademic Press Incen_US
dc.relation.ispartofJournal of Algebraen_US
dc.relation.publicationcategoryMakale - Uluslararası Hakemli Dergi - Kurum Öğretim Elemanıen_US
dc.rightsinfo:eu-repo/semantics/openAccessen_US
dc.subjectVariety Of Algebrasen_US
dc.subjectFinitely Determined Algebraen_US
dc.subjectExistentially Closed Algebraen_US
dc.subjectWord Problemen_US
dc.subjectCenter-By-Metabelian Groupen_US
dc.subjectAssociative Ringen_US
dc.titleFinitely determined members of varieties of groups and rings
dc.typeArticle

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