Mackey Functors with Chain Conditions

AbstractLetAbe a Green functor for a finite groupGand letMbe a leftA-module.Mis callednoetherian (artinian)if the lattice of all submodules ofMsatisfies ACC (respectively DCC).Mis calledtotally noetherian (totally artinian)ifMXis noetherian (respectively artinian) for every finiteG-setX. In this paper it is shown that the totally noetherian (totally artinian) leftA-modules behave in the same way as the classical noetherian (artinian) modules. The left noetherian (left artinian) Green functors are less well behaved but some properties from classical algebra still hold. It is shown that Cohen's theorem holds in this setting. Our main result shows that if a commutative Green functorAis artinian, then the ringsA(H)/Jac(A(H)) are semisimple artinian for all subgroupsHofG