The Hilbert basis theorem revisited (Q921104)

An analogue of the Hilbert Basis Theorem for commutative monoids in Hilbert categories is established. A Hilbert category is a (symmetric) tensored abelian category with countable coproducts in which the functors \(-\otimes C\) and \(C\otimes -\) preserve all countable colimits. As a special case, a necessary and sufficient condition is given for the free commutative monoid on an object of a Hilbert category to be noetherian. An example is presented to distinguish this condition from the more obvious condition for the generating object to be noetherian.