We investigate the structure of locally finite profinite rings. We classify (Jacobson-) semisimple locally finite profinite rings as products of complete matrix rings of bounded cardinality over finite fields, and we prove that the Jacobson radical of any locally finite profinite ring is nil of finite nilexponent. Our results apply to the context of small compact G-rings, where we also obtain a...

A k-linear triangulated category A is called locally finite provided ∑ X∈indA dimk HomA(X,Y ) < ∞ for any indecomposable object Y in A. It has Auslander–Reiten triangles. In this paper, we show that if a (connected) triangulated category has Auslander–Reiten triangles and contains loops, then its Auslander–Reiten quiver is of the form L̂n: · · · · ̂ n n− 1 2 1 By using this, we prove that the Aus...

A 1nite graph is said to be locally-quasiprimitive relative to a subgroup G of automorphisms if, for all vertices , the stabiliser in G of is quasiprimitive on the set of vertices adjacent to . (A permutation group is said to be quasiprimitive if all of its non-trivial normal subgroups are transitive.) The graph theoretic condition of local quasiprimitivity is strictly weaker than the condition...