Generating subdirect products

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, · · · , A_n for a Noetherian ring K such that the projection of C onto any A_i × A_j has finite co-rank in A_i × A_j. Then C is finitely generated (respectivley, finitely presented) if and only if all A_i 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