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Generating subdirect products

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Abstract

We study conditions under which subdirect products of various types of algebraic structures are finitely generated or finitely presented. In the case of two factors, we prove general results for arbitrary congruence per-mutable varieties, which generalise previously known results for groups, and which apply to modules, rings, K-algebras and loops. For instance, if C is a fiber product of A and B over a common quotient D, and if A, B and D are finitely presented, then C is finitely generated. For subdirect products of more than two factors we establish a general connection with projections on pairs of factors and higher commutators. More detailed results are provided for groups, loops, rings and K-algebras. In particular, let C be a subdirect product o fK-algebras A1, · · · , An for a Noetherian ring K such that the projection of C onto any Ai × Aj has finite co-rank in Ai × Aj. Then C is finitely generated (resp. finitely presented) if and only if all Ai are finitely generated (resp. finitely presented). Finally, examples of semigroups and lattices are provided which indicate further complications as one ventures beyond congruence permutable varieties
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Original languageEnglish
Pages (from-to)404-424
JournalJournal of the London Mathematical Society
Volume100
Issue number2
Early online date9 Apr 2019
DOIs
Publication statusPublished - 1 Oct 2019

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