Minimal order morphisms (Q687540)

A ring morphism \(f\) is minimal if it is injective and non-surjective and into for every decomposition \(f=g \circ h\) in injective ring morphisms, \(h\) or \(g\) is an isomorphism. The author studies morphisms between two orders and their decomposition into minimal morphisms. He finds a kind of a canonical decomposition \(f=f_ 1\circ f_ 2 \circ f_ 3\) \((f_ 1\) is composed only from inert morphisms, \(f_ 2\) is decomposed and \(f_ 3\) is ramified). It is shown that a decomposition of an morphism into minimal morphisms need not be unique -- the length of the decomposition is not constant for a given morphism.