Generating subdirect products
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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 permutable varieties, which generalize 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 (respectivley, finitely presented) if and only if all Ai are finitely generated (respectively, finitely presented). Finally, examples of semigroups and lattices are provided which indicate further complications as one ventures beyond congruence permutable varieties
Mayr , P & Ruskuc , N 2019 , ' Generating subdirect products ' , Journal of the London Mathematical Society , vol. 100 , no. 2 , pp. 404-424 . https://doi.org/10.1112/jlms.12221
Journal of the London Mathematical Society
Copyright © 2019 London Mathematical Society. This work has been made available online in accordance with the publisher's policies. This is the author created accepted version manuscript following peer review and as such may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1112/jlms.12221
DescriptionThe first author was supported by the National Science Foundation under Grant No. DMS 1500254.
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