MULTIPLICATION RINGS AS RINGS IN WHICH IDEALS WITH PRIME RADICAL ARE PRIMARY

A commutative ring R is called an AM-ring (for allgemeine multiplikationsring) if whenever A and B are ideals of R with A properly contained in B, then there is an ideal C of R such that A = BC. An AM-ring R in which RA = A for each ideal A of R is called a multiplication ring. Krull introduced the notion of a multiplication ring in [11], [13]. Akizuki is responsible for the more general concept of an AM-ring in [1], but Mori has developed most of the structure theory for such rings in [14], [15], [16], [17], and [18]. An important property of an AM-ring R is that R satisfies what Gilmer called condition (*) in [ 7 ] and [8 ]: An ideal of R with prime radical is primary. In ? 1, new results concerning rings in which (*) holds are given. These are applied to obtain structure theorems for AM-rings in ? 2. In [10, p. 737], Krull introduced the notion of the kernel of an ideal A in a commutative ring R, which is defined thusly: if Pa } is the collection of minimal prime ideals of A, then by an isolated primary component of A belonging to Pa we mean the intersection Qa of all Pay-primary ideals which contain A. The kernel of A is the intersection of all Qa's. Mori considered rings in which every ideal is equal to its kernel in [ 16] and [ 17 ]. In ? 1, it is shown that a ring R satisfies (*) if and only if every ideal of R is equal to its kernel. In ?2 of this paper, we consider rings R satisfying condition (F): If A and P are ideals of R such that P is prime and A is properly contained in P, there is an ideal B such that A = PB. Theorems 12 and 13 show that such a ring is an AM-ring. This generalizes a result proved by Mott in [19] for rings with unit. This theorem might be compared with the result of Cohen [3, Theorem 2, p. 29], that a ring in which every prime ideal is finitely generated is a Noetherian ring and to the theorem of Nakano in [ 20, p. 234] which states that every nonzero ideal of an integral domain D with unit is invertible provided every nonzero prime ideal of D is invertible. In addition, several new sets of necessary and sufficient conditions that a ring R be an AM-ring are given in ? 2. An equally significant aspect of ? 2 in our eyes is that in the process of proving Theorem 12, many of the known results concerning AM-rings are proved in a way we feel is clearer and more straight-