## Abstract

We say that a subring R_{0} of a ring R is semi-invariant if R_{0} is the ring of invariants in R under some set of ring endomorphisms of some ring containing R.We show that R_{0} is semi-invariant if and only if there is a ring S ⊇ R and a set X ⊆ S such that R_{0} = Cent_{R}(X): = {r ∈ R: x r = r x ∀ x ∈ X}; in particular, centralizers of subsets of R are semi-invariant subrings.We prove that a semi-invariant subring R_{0} of a semiprimary (resp. right perfect) ring R is again semiprimary (resp. right perfect) and satisfies Jac(R0)n⊆Jac(R) for some n∈N. This result holds for other families of semiperfect rings, but the semiperfect analogue fails in general. In order to overcome this, we specialize to Hausdorff linearly topologized rings and consider topologically semi-invariant subrings. This enables us to show that any topologically semi-invariant subring (e.g. a centralizer of a subset) of a semiperfect ring that can be endowed with a "good" topology (e.g. an inverse limit of semiprimary rings) is semiperfect.Among the applications: (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 End(M) is semiperfect, hence M has a Krull-Schmidt decomposition. (This generalizes results of Bjork and Rowen; see Björk (1971) [5], Rowen (1986, 1987) [23,24].) (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 End(M_{R}) is semiprimary (resp. right perfect), then End(M_{S}) is semiprimary (resp. right perfect).

Original language | English |
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Pages (from-to) | 103-132 |

Number of pages | 30 |

Journal | Journal of Algebra |

Volume | 378 |

DOIs | |

State | Published - 5 Mar 2013 |

Externally published | Yes |

### Bibliographical note

Funding Information:This research was partially supported by an Israel–US BSF grant #2010/149. E-mail address: [email protected].

## Keywords

- Centralizers
- Krull-Schmidt decomposition
- Linearly topologized rings
- Rationally closed subring
- Ring theory
- Rings of invariants
- Semi-invariant subrings
- Semiperfect ring

## ASJC Scopus subject areas

- Algebra and Number Theory