It is well known that the direct product of two groups is finitely generated if and only if both factors are finitely generated. But is this a result about groups, classical structures with a group operation (e.g., rings, associative algebras,...), semigroups, algebras with neutral element, or true for all algebraic structures? We use universal algebra to investigate this question and corresponding results for finite presentability and residual finiteness of algebras.
The Unbearable Lightness of Direct Products