PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 130, Number 6, Pages 1573–1580 S 0002-9939(01)06231-1 Article electronically published on October 24, 2001 THE HOMOGENEOUS SPECTRUM OF A GRADED COMMUTATIVE RING WILLIAM HEINZER AND MOSHE ROITMAN (Communicated by Wolmer V. Vasconcelos) Abstract. Suppose Γ is a torsion-free cancellative commutative monoid for which the group of quotients is finitely generated. We prove that the spectrum of a Γ-graded commutative ring is Noetherian if its homogeneous spectrum is Noetherian, thus answering a question of David Rush. Suppose A is a com- mutative ring having Noetherian spectrum. We determine conditions in order that the monoid ring A[Γ] have Noetherian spectrum. If rank Γ ≤ 2, we show that A[Γ] has Noetherian spectrum, while for each n ≥ 3 we establish existence of an example where the homogeneous spectrum of A[Γ] is not Noetherian. 0. Introduction All rings we consider are assumed to be nonzero, commutative and with unity. All the monoids are assumed to be torsion-free cancellative commutative monoids. Let Γ be a monoid such that the group of quotients G of Γ is finitely generated, and let R =  γ∈Γ R γ be a commutative Γ-graded ring. A goal of this paper is to answer in the affirmative a question mentioned to one of us by David Rush as to whether Spec R is necessarily Noetherian provided the homogeneous spectrum, h-Spec R, is Noetherian. If I is an ideal of a ring R, we let rad(I ) denote the radical of I , that is rad(I )= {r ∈ R : r n ∈ I for some positive integer n}. We say that I is a radical ideal if rad(I )= I . A subset S of the ideal I generates I up to radical if rad(I ) = rad(SR). The ideal I is radically finite if it is generated up to radical by a finite set. We recall that a ring R is said to have Noetherian spectrum if the set Spec R of prime ideals of R with the Zariski topology satisfies the descending chain condition on closed subsets. In ideal-theoretic terminology, R has Noetherian spectrum if and only if R satisfies the ascending chain condition (a.c.c.) on radical ideals. Thus a Noetherian ring has Noetherian spectrum and each ring having only finitely many prime ideals has Noetherian spectrum. As is shown in [8, Prop. 2.1], Spec R is Noetherian if and only if each ideal of R is radically finite. It is well known that R has Noetherian spectrum if and only if R satisfies the two properties: (i) a.c.c. on prime ideals, and (ii) every ideal of R has only finitely many minimal prime ideals [6], [3, Theorem 88, page 59 and Ex. 25, page 65]. Received by the editors September 20, 2000 and, in revised form, December 13, 2000. 1991 Mathematics Subject Classification. Primary 13A15, 13E99. Key words and phrases. Graded ring, homogeneous spectrum, Noetherian spectrum, torsion- free cancellative commutative monoid. This work was prepared while the second author enjoyed the hospitality of Purdue University. c 2001 American Mathematical Society 1573