Rings of invariants under endomorphisms

Wed, 01/02/2012 - 10:30

The classical scenario in the algebraic theory of invariants is where a group G of automorphisms acts on a ring R. Working in a more general setting, where G need not be a group, I will discuss properties of R which are inherited by the ring of invariants R^G, focusing on cases when R is "almost" semisimple Artinian. 
In particular, if R is semiprimary (resp. left/right perfect; semilocal complete) then so is the invariant ring R^G for any set G of endomorphisms of R. However, that R is artinian or semiperfect need not imply this property for R^G, even when G is a finite group with an inner action. (Examples will be presented if time permits.) The former result actually holds in a more general context: Let S be a ring containing R and let G be a set of endomorphisms of S, then the ring R^G of G-invariant elements inside R inherits from R the properties: being semiprimary, being left (resp. right) perfect.
As easy corollaries, we get that if R is a subring of a ring S, then the centralizer in R of any subset of S inherits the property of being semiprimary or left perfect from R. Better still, the centralizer in R of a set of invertible elements in R inherits the property of being semilocal-complete.
Similarly, assume S is a ring containing R and let M be a right S-module. Then, that End_R(M) is semiprimary (resp. left/right perfect) implies that End_S(M) is. 
All ring-theoretic notions will be defined.