The structure of prime homogeneous ideals in graded amalgamated algebra along an ideal

Abstract

Let [Formula: see text] and [Formula: see text] be two commutative rings graded by an arbitrary commutative monoid [Formula: see text], [Formula: see text] be a homogeneous ideal of [Formula: see text], and [Formula: see text] be a graded ring homomorphism. The amalgamation of [Formula: see text] with [Formula: see text] along [Formula: see text] with respect to [Formula: see text] is a graded ring, where [Formula: see text] for each [Formula: see text]. In this paper, we first completely describe the prime homogeneous spectrum of the graded amalgamated algebra rooted in the form of the pullback. Then we answer the question of when the graded amalgamation [Formula: see text] is a graded von Neumann regular ring.