Semi-Invariant Subrings

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We say that a subring R₀ of a ring R is semi-invariant if R₀ is the ring of invariants in R under some set of ring endomorphisms of some ring containing R. We show that R₀ is semi-invariant if and only if there is a ring S R and a set X S such that R₀=ᵣ(X):=r∈ R xr=rx ∀ x∈ X; in particular, centralizers of subsets of R are semi-invariant subrings. We prove various properties of semi-invariant subrings and show how they can be used for various applications including: (1) The center of a semiprimary (resp. right perfect) ring is semiprimary (resp. right perfect). (2) If M is a finitely presented module over a "good" semiperfect ring (e.g. an inverse limit of semiprimary rings), then (M) is semiperfect, hence M has a Krull-Schmidt decomposition. (This generalizes results of Bjork and Rowen). (3) If ρ is a representation of a monoid or a ring over a module with a "good" semiperfect endomorphism ring (in the sense of (2)), then ρ has a Krull-Schmidt decomposition. (4) If S is a "good" commutative semiperfect ring and R is an S-algebra that is f.p.as an S-module, then R is semiperfect. (5) Let R S be rings and let M be a right S-module. If (Mᵣ) is semiprimary (resp. right perfect), then (Mₛ) is semiprimary (resp. right perfect).