Graded pseudo-almost valuation domains

Abstract

Let [Formula: see text] be a nonzero commutative cancellative monoid (written additively), [Formula: see text] a [Formula: see text]-graded integral domain, and [Formula: see text] the saturated multiplicative set of nonzero homogeneous elements of [Formula: see text]. A homogeneous prime ideal [Formula: see text] of [Formula: see text] is said to be a pseudo-strongly homogeneous prime ideal if for each homogeneous elements [Formula: see text] whenever [Formula: see text], then there exists a positive integer [Formula: see text], such that either [Formula: see text] or [Formula: see text]. A graded integral domain [Formula: see text] is said to be a graded pseudo-almost valuation domain (gr-PAVD) if each homogeneous prime ideal of [Formula: see text] is a pseudo-strongly homogeneous prime ideal. We study the prime ideal- and ring-theoretic properties and overrings of gr-PAVDs. We also study the gr-PAVD property in pullback of graded domains and give various examples of these domains.