Fraction-dense algebras and spaces

Anthony W. Hager, Jorge Martinez
1993 Canadian Journal of Mathematics - Journal Canadien de Mathematiques  
A fraction-dense (semi-prime) commutative ring A with 1 is one for which the classical quotient ring is rigid in its maximal quotient ring. The fractiondense/-rings are characterized as those for which the space of minimal prime ideals is compact and extremally disconnected. For archimedean lattice-ordered groups with this property it is shown that the Dedekind and order completion coincide. Fractiondense spaces are defined as those for which C(X) is fraction-dense. If X is compact, then this
more » ... ompact, then this notion is equivalent to the coincidence of the absolute of X and its quasi-/ 7 cover. /?-embeddings of Tychonoff spaces are re-introduced and examined in the context of fraction-density. Introduction. Fraction-dense algebras arise naturally in the consideration of quotient rings, and they give rise to an interesting class of topological spaces. For archimedean Riesz spaces Huijsmans and de Pagter have introduced a similar concept, in [HP], which they call almost Dedekind completeness. In fact, what we shall later call an absolute l-group coincides, in the context of uniformly complete Riesz spaces, with an almost Dedekind complete Riesz space. There is considerable overlap in the algebraic parts between this article and [HP]. We shall, however not attempt to reconcile their terminology with ours. Unless further qualified, every ring in this exposition will be commutative, possess an identity, and also be semi-prime, in the sense that there are no non-zero nilpotent elements. An/-ring is a lattice-ordered ring in which a A b -0 implies that a A be = 0 for each c > 0. In ZFC this is equivalent to requiring that the lattice-ordered ring be a subdirect product of totally ordered rings. Likewise, all topological spaces are assumed to be Tychonoff, unless the contrary is expressly stated. Recall that a Hausdorff space is Tychonoff if the cozero-sets (of real-valued continuous functions) form a base for the topology. All lattice-ordered groups in this article are abelian. Our standard references for this theory are [AF] and [BKW]. Suppose that A is an/-ring; then qA stands for its classical ring of quotients and QA for its maximal ring of quotients. qA should be familiar to the reader; however, let us recall some properties of QA. (For further reading we refer the reader to [La], [Ba] and [M].) First of all, the term "ring of quotients" should be interpreted as follows: assume that A is a subring of the ring B; we say that B is a ring of quotients of A if for each pair b], Z?2, with Z?2 7^ 0, there exists an a G A such that ab\ and abi both belong to A and ab2 7^ 0. Each ring then has a (unique) maximal ring of quotients; in [La] the subject
doi:10.4153/cjm-1993-054-6 fatcat:4i2ykqbeqbbslas2apv2snkj3e