Right-bounded factors in an LCM domain

Abstract

A right-bounded factor is an element in a ring that generates a right ideal which contains a nonzero two-sided ideal. Right-bounded factors in an LCM domain are considered as a generalization of the theory of two-sided bounded factors in an atomic 2-fir, that is, a weak Bezout domain satisfying the acc and dcc for left factors. Although some elementary properties are valid in a more general context most of the main results are obtained for an LCM domain satisfying (M) and the dcc for left factors; the condition (M) is imposed to insure that prime factorizations are unique in an appropriate sense. The right bound b* of a right bounded element b is considered in general, then in case b is a prime, and finally in case b is indecomposable. The effect of assuming that right bounds are two-sided is also considered. 0. Introduction. The theory of bounded factors in a principal ideal domain is well established [11]. More recently, this was generalized to 2-firs (i.e. weak Bezout domains) satisfying the acc and dcc for left factors [6]. Our purpose here is twofold: (i) to study right-bounded factors, and (ii) to carry this out in the more general context of right LCM domains (intersection of any two principal right ideals is principal), a class of rings which was described in [2] and [3]. It was shown in [2] that for right LCM domains satisfying an additional mild hypothesis (M) factorization into primes is unique up to order of factors and projective factors. In ? I we collect this and other related facts that will be needed. Right-bounded factors in a ring R are considered in ?2; their right bounds exist if R is a complete right LCM domain (intersection of any collection of principal right ideals is principal). In ?3 we consider right-bounded primes. The right bound p* of a prime p is described in some detail. For example, it is shown that if R is an LCM domain satisfying (M) and the dcc for left factors then p* can be factored into primes that are projective (in fact transposed) to p. The possibility of factoring p* which is right invariant into further right invariant factors is also discussed. In ?4 we consider right bounded elements Received by the editors September 9, 1973. AMS (MOS) subject classifications (1970). Primary 16A02, 16A04, 16A08.