A ring (resp. module) is called Noetherian if every ascending chain of ideals (submodules) becomes eventually stationary. According to the Hilbert's Basis Theorem every finitely generated module over a finitely generated K-algebra is Noetherian (see [Rei88], [Kre00] or others). Please consider to think about Hilbert's (projective) Nullstellensatz

Here the definition of the radical ideal slightly differs from the one in the section called “Group Rings and Algebras”. All this is quite important for the XiStrat™ project.

Due to Noether and moreover since finite groups are (geometrically) reductive, their invariant rings are finitely generated and graded. The Hilbert-Poincaré series encodes the dimensions of the ideals of generators of given degree. A graded algebra is Cohen-Macaulay if and only if it is free as a module over a subalgebra generated by a homogeneous system of parameters. A polynomial ring is Cohen-Macaulay. For a finite group G, the invariant ring is polynomial iff (if and only if) G is a generalized reflection group.

The theory depends on what representation is taken. V denotes a faithful irreducible representation and K[V] G the invariant ring of G in what follows. If char(K) does not divide the group order, then K[V] G is Cohen-Macaulay. In the modular case the invariant ring is in general not that nice, and the Molien series gives only the so called extended Hilbert series H K[V] G t .

gap>ctl:=CharacterTable(AlternatingGroup(5));

gap>psi:=First(Irr(ctl),x->Degree(x)=3);
Character( CharacterTable( Alt( [ 1 .. 5 ] ) ), 
[ 3, -1, 0, -E(5)^2-E(5)^3, -E(5)-E(5)^4 ] )
gap>MolienSeries(psi);
( 1-z^2-z^3+z^6+z^7-z^9 ) / ( (1-z^5)*(1-z^3)*(1-z^2)^2 )

How about the inverse problem: Given some invariant ring R, construct a group such that R is the group's invariant ring.

As a side note we ask whether perhaps the knot invariants have to do with the invariant ring of a group?