Abstract
Dobbs characterized 1-split domains. Considering the total integral closure of Hochster, we generalize the definitions and results of Dobbs to arbitrary reduced rings. If R is a reduced ring, then R is 1-split if and only if R is a Baer ring and R P is a henselian domain for each P ∈ Spec(R). The spectra of the local rings of a 1-split ring are complete lattices. If R is 1-split and f : R → R′ is a ring morphism which is either integral or quasi-finite, then f has the strong going-between property. Using Raphael's definition of algebraic extensions of rings, we get that a reduced ring R is absolutely injective if and only if R is 1-split and extensionally injective. Universally going-down weak Baer rings are characterized by means of their total integral closures. Their class is stable under étale morphisms.
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