Valuative rings

Abstract

For a pair of commutative rings [Formula: see text] with the same identity, [Formula: see text] is said to be [Formula: see text]-valuative if for each nonzero [Formula: see text] at least one of the extensions [Formula: see text] and [Formula: see text] has no proper subrings. There are weak and strong versions: [Formula: see text] is weakly [Formula: see text]-valuative if for each pair [Formula: see text] such that [Formula: see text], at least one of [Formula: see text] and [Formula: see text] has no proper intermediate rings; [Formula: see text] is strongly [Formula: see text]-valuative if for each nonzero [Formula: see text] at least one of the extensions [Formula: see text] and [Formula: see text] has no proper intermediate rings. Here, we consider the special case that [Formula: see text] is the total quotient ring. Examples are provided to show that even in this setting there are [Formula: see text]-weakly valuative rings that are not [Formula: see text]-valuative, and [Formula: see text]-valuative rings that are not strongly [Formula: see text]-valuative. In contrast, if [Formula: see text] is weakly additively regular, then it is weakly [Formula: see text]-valuative if and only if it is strongly [Formula: see text]-valuative.