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 co-rank 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.