Generating subdirect products
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 permutable 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 of $K$algebras $A_1,\dots,A_n$ for a Noetherian ring $K$ such that the projection of $C$ onto any $A_i\times A_j$ has finite corank in $A_i\times A_j$. Then $C$ is finitely generated (resp. finitely presented) if and only if all $A_i$ 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.
 Publication:

arXiv eprints
 Pub Date:
 February 2018
 DOI:
 10.48550/arXiv.1802.09325
 arXiv:
 arXiv:1802.09325
 Bibcode:
 2018arXiv180209325M
 Keywords:

 Mathematics  Rings and Algebras;
 Mathematics  Group Theory;
 08B26;
 08B10;
 16S15;
 20F05
 EPrint:
 doi:10.1112/jlms.12221