The Prüfer rank, also known as the Prüfer order, is a concept from the field of algebraic topology and algebraic K-theory that applies to modules, particularly in relation to Prüfer domains. It is a measure of the "size" of a module, similar to the rank of a vector space, but adapted for module theory.
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The Prüfer rank bounds the number of generators of a group needed for each finitely generated subgroup, without requiring the ambient group to be finitely generated. The additive group has Prüfer rank one: any finitely generated subgroup has a common denominator and is cyclic, but no finite collection generates all rational numbers. Prüfer rank cannot increase under taking subgroups or quotients; for quotients, lift a finite generating set and apply the rank bound to the subgroup generated by its lifts.