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Factorization in Integral Domains, II View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 152,78-93 (1992) Factorization in Integral Domains, II D. D. ANDERSON Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242 DAVID F. ANDERSON* Department of Mathematics, The Unioersity of Tennessee, Knoxville, Tennessee 37996 AND MUHAMMAD ZAFRULLAH Department of Mathematics, Winthrop College, Rock Hill, South Carolina 29733 Communicated by Richard G. Swan Received June 8. 1990 In this paper, we study several factorization properties in an integral domain which are weaker than unique factorization. We study how these properties behave under localization and directed unions. 0 1992 Academic Press, Inc. In this paper, we continue our investigation begun in [l] of various factorization properties in an integral domain which are weaker than unique factorization. Section 1 introduces material on inert extensions and splitting multiplicative sets. In Section 2, we study how these factorization properties behave under localization. Special attention is paid to multi- plicative sets generated by primes. Section 3 consists of “Nagata-type” theorems about these properties. That is, if the localization of a domain at a multiplicative set generated by primes satisfies a certain ring-theoretic property, does the domain satisfy that property? In the fourth section, we consider Nagata-type theorems for root closure, seminormality, integral * Supported in part by a National Security Agency Grant. 78 OO21-8693/92$5.00 Copyright 0 1992 by Academic Press, Inc. All righrs of reproduction in any form reserved. FACTORIZATION IN INTEGRAL DOMAINS, II 79 closure: and complete integral closure. In the final section, we investigate how these factorization properties behave under directed unions. We first recall the various factorization properties in addition to unique factorization which we will study here. Throughout, let R be an integral domain with quotient field K. Following Cohn [S], we say that R is atomic if each nonzero nonunit of R is a product of a finite number of irreducible elements (atoms) of R. We say that R satisfies the ascending chain condition on principal ideals (ACCP) if there does not exist an infinite strictly ascending chain of principal integral ideals of R. The domain R is a bounded factorization domain (BFD) if R is atomic and for each nonzero nonunit x E R there is a bound on the lengths of factorizations of x into products of irreducible elements (equivalently, there is a bound on the lengths of chains of principal integral ideals starting at xR). Following Zaks [17], we say that R is a half-factorial domain (HFD) if R is atomic and each factorization of a nonzero nonunit of R into a product of irreducible elements has the same length. Following Grams and Warner [ 121, we say that the domain R is an idf-domain (for irreducible-divisor- finite) if each nonzero element of R has at most a finite number of nonassociate irreducible divisors. An atomic idf-domain will be called a finite factorization domain (FFD); these are precisely the domains in which each nonzero nonunit has only a finite number of nonassociate divisors (and hence, only a finite number of factorizations up to order and associates). BFDs and FFDs were introduced in [ I]. These factorization properties have also been studied in [2]. In general, HFD Y B UFD * FFD 3 BFD a ACCP * atomic \ u idf-domain Examples given in [l] show that no other implications are possible. General references for any undefined terminology or notation are [7,9, 131. For an integral domain R, R* is its set of nonzero elements, U(R) its group of units, R’ its integral closure, and i? its complete integral closure. By an ideal, we always mean an integral ideal. For nonzero a, b E R, (a, b) = 1 means that a and b have no nonunit common factors. We also make the two harmless assumptions that all our multiplicative sets do not contain 0 and are saturated. A multiplicative set S is generated by Tc R if S = {ut, . t, 1u E U(R) and tj E r>. Throughout, Z, Q, and R denote the integers, rational numbers, and real numbers, respectively. 80 ANDERSON, ANDERSON, AND ZAFRULLAH 1. INERT EXTENSIONS AND SPLITTING MULTIPLICATIVE SETS Following Cohn [S], we say that an extension of rings A c B is an inert extension if whenever xy E A for nonzero x, y E B, then XU, yu- ’ E A for some u E U(B). Our first lemma is LEMMA 1.1. Let A c B be an inert extension of integral domains. Then an irreducible element of A is either irreducible or a unit in B. ProoJ: Let a E A be irreducible and suppose that a is not a unit in B. If a = XJJ for x, ye B, then xu, yu-i E A for some u E U(B). Hence a = (xu)(yu-‘) in A implies that either xu or yu-’ is a unit in A. Thus either x or y is a unit in B. Hence a is irreducible in B. 1 Easy examples show that either case may occur in Lemma 1.1. Also, an a E A may be irreducible in B, but not irreducible in A. However, if A c B is an inert extension and U(B) n A = U(A), then an a E A is irreducible in A if and only if it is irreducible in B. It is easily seen that none of our factorization properties need ascend or descend for an inert extension A c B of integral domains. We next define a special type of multiplicative set. A (saturated) multi- plicative subset S of R is a splitting multiplicative set if for each XE R, x = as for some c1E R and s E S such that aR n tR = atR for all t E S. (Thus a E S if and only if a E U(R).) Similar types of multiplicative sets have been studied in [lo, Sect. 3; 14, Sect. 4]. (This should not be confused with splittable sets as defined by Zaks in [17, 181.) We first have (cf. [14, Proposition 4.11) LEMMA 1.2. Let R be an integral domain and S a multiplicative set of R. Then the following statements are equivalent for x, y E R: (a) xR, n R = yR. (b) x = ys for some s E S and yR n tR = ytR for all t E S. ProoJ: (a) => (b) We may assume that x and y are nonzero. We have x = ys for some SE R since XE xR,n R = yR. Also, y = x(r/t’) for some r E R and t’E S. Thus xr = yt’, so sr = t’ E S. Hence s E S since S is saturated. We next show that yR n tR = ytR for all t E S. The “3” inclu- sion is clear. Conversely, let z E yR n tR. Then z = ya = tb for some a, b E R. Thus yas = tbs, and hence ax = tbs. Thus b = x(a/(st)) ~xRs n R = yR, so b = yc for some c E R. Hence z = ytc E ytR. Thus the “c” inclusion holds and we have equality. (b) * (a) Let x = ys with s E S and yR n tR = ytR for all t E S. Then xR, = yR,, so we need only show that yR, n R = yR. The “3” inclusion FACTORIZATION IN INTEGRAL DOMAINS, II 81 is clear. Conversely, let z E yR, n R. Thus z = y(r/t) for some YE R and t E S. Hence tz = yr E yR n tR = ytR, so z E yR. Hence the “c” inclusion holds and we have equality. 1 COROLLARY 1.3. A multiplicative set S of an integral domain R is a splitting multiplicative set if and only if principal ideals of R, intersect to principal ideals of R. COROLLARY 1.4. Let R be an integral domain and S a splitting multi- plicative set of R. Let XE R be nonzero and x = as uith aE R, s E S, and aRntR=atRfor all tES. (a) If x = a’s’ irith a’ E R, s’ E S, and a’R A tR = a’tR for all t E S, then a and a’ are associates and s and s’ are associates. (b) If y= bt tvith be R, tES, and bRn t’R= bt’R for aN t’ ES, then abRn t’R= abt’R for all t’E S. Hence the decomposition for xy is (ab)(st). (cl x is prime (resp., irreducible) in R, if and only if a is prime (resp., irreducible) in R. (d) Each prime (resp., irreducible) element in R, is an associate in R, of a prime (resp., irreducible) element in R. ProojI (a) This follows directly from Lemma 1.2. (b) This follows because abt’R = b(aR n t’R) = abR n bt’R = abR n bR n t’R = abR r, t’R. (c) If x is prime in R,, then a is prime in R since aR = xR,n R. The converse is clear. If a is irreducible in R, then x is irreducible in R, by Lemma 1.1 (and Proposition 1.5) since SE U(R,). Conversely, if a= ala2 with neither factor a unit in R, then a, and hence x, is not irreducible in R,. (d) This follows easily from (c). 1 In general, localization need not yield an inert extension R c R,. IIowever, we next show that this extension is inert when S is a splitting multiplicative set.
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