Monoid Domain Constructions of Antimatter Domains

An integral domain without irreducible elements is called an antimatter domain. We give some monoid domain constructions of antimatter domains. Among other things, we show that if D is a GCD domain with quotient …eld K that is algebraically closed, real closed, or perfect of characteristic p > 0, then the monoid domain D[X;Q+] is an antimatter GCD domain. We also show that a GCD domain D is antimatter if and only if P 1 = D for each maximal t-ideal P of D. Let D be an integral domain with quotient …eld K. By an irreducible element or atom of D we mean a nonunit x 2 D = D f0g such that x = uv, u; v 2 D, implies u or v is a unit. The domain D is atomic if each nonzero nonunit of D is expressible as a …nite product of atoms. However, it may happen that a domain does not have any atoms. Such domains, called antimatter domains, were introduced by Coykendall, Dobbs, and Mullins [5]. A somewhat obvious example of an antimatter domain is a valuation domain whose maximal ideal is not principal [5, Proposition 1]. Another example is a …eld which, ironically, is also an example of an atomic domain. It is patent that if D is an antimatter domain, or any integral domain for that matter, then D[X] is not antimatter, as X + r is an atom in D[X] for all r 2 D. On the other hand, the monoid domain C[X;Q], where Q is the monoid of nonnegative rationals under addition, is an antimatter domain (Theorem 1). But Q[X;Q] is not antimatter as X 2 is irreducible. (If X 2 properly factors in Q[X;Q], then X 2 properly factors in some Q[X] since Q is locally cyclic (that is, each …nitely generated submonoid of Q is contained in a cyclic submonoid of Q). But by Eisenstein’s Criterion, X 2 = (X) 2 is irreducible in Q[X]). The purpose of this paper is to explore the following question. For an integral domain D and torsionless cancellative monoid S (always written additively), when is the monoid domain D[X;S] antimatter? Certainly, if D[X;S] is antimatter, then D and S must be antimatter (a monoid S is antimatter if it has no atoms where atoms are de…ned in the obvious way). However, as both Q and (Q;+) are antimatter while Q[X;Q] is not, the converse is false. In this note we show that if D is an antimatter GCD domain with quotient …eld K algebraically closed, real closed, or perfect of characteristic p > 0, (Theorems 1, 2, and 5), then D[X;Q] is an antimatter domain. Our standard references are [6], [7], and [10]. In the case where D = K is an algebraically closed or real closed …eld, we can show that D[X;S] is antimatter in slightly more generality than the case S = (Q;+). Let us call a monoid S pure if (1) S is (order-isomorphic to) a submonoid