Mackey functors with chain conditions (Q1270391)

Let \(A\) be a Green functor for a finite group \(G\) over a commutative ring \(R\). The Jacobson radical \(J(A)\) can be defined as the largest ideal \(I\) of \(A\) such that \(I(H)\leq J(A(H))\) for \(H\leq G\). A left \(A\)-module \(M\) is called Artinian (Noetherian) if it satisfies the descending (ascending) chain condition on submodules. The author shows: Every left Artinian Green functor \(A\) is also left Noetherian, \(J(A)\) is nilpotent, and there are only finitely many isomorphism classes of simple \(A\)-modules. A Green functor \(A\) is called commutative if \(A(H)\) is commutative for \(H\leq G\). The author proves that in this case \(A\) is Artinian with \(J(A)=0\) if and only if \(A(H)\) is Artinian with \(J(A(H))=0\) for \(H\leq G\). He also illustrates that several results on chain conditions in ring theory do not carry over directly to modules over Green functors. Some of these difficulties can be avoided by dealing with a smaller class of modules called totally Artinian (Noetherian).