Almost-Dedekind rings

E. W. Johnson
1994 Glasgow Mathematical Journal  
Throughout we assume all rings are commutative with identity. We denote the lattice of ideals of a ring R by L(R), and we denote by L(R)* the subposet L(R) -R. A classical result of commutative ring theory is the characterization of a Dedekind domain as an integral domain R in which every element of L(R)* is a product of prime ideals (see Mori [5] for a history). This result has been generalized in a number of ways. In particular, rings which are not necessarily domains but which otherwise
more » ... hich otherwise satisfy the hypotheses (i.e. general ZPI-rings) have been widely studied (see, for example, Gilmer [3]), as have rings in which only the principal ideals are assumed to satisfy the hypothesis (i.e. ^-rings). General ZPI-rings and ^-rings can both be thought of as "almost Dedekind". In both cases, one gets a representation as the finite direct product of integral domains of the same type (Dedekind domains in the first case, ^-domains in the second case) and quotients of discrete (rank one) valuation rings (i.e. special principal ideal rings-or SPIRS as they have come to be called). Note that ZPI-rings are rings in which every ideal in L(R)* satisfies the "product of prime ideals" condition, whereas only the principal ideals of a ;r-ring are assumed to satisfy this condition. This naturally raises consideration of rings in which every ideal of L(R)* generated by n elements is a product of prime ideals. Any UFD is a Jr-ring; so a jr-ring need not be a general ZPI-ring. In this regard, Levitz [4, 5] has obtained the very interesting result that wrings are the single exception. If every doubly generated ideal in L(R)* is the product of prime ideals, then every ideal in L(R)* is. Butts and Gilmer [3] have characterized ZPI-rings in a somewhat different manner. They have shown that ZPI-rings are characterized by the property that every ideal in L(R)* is a finite intersection of powers of prime ideals. In this paper, we obtain the analogue of Levitz's theorem for the Butts-Gilmer characterization of general ZPI-rings. That is, we show that, once again, two elements suffice: if R is a ring in which every double generated ideal in L(R)* is the intersection of powers of prime ideals, then every ideal in L(R)* is. For convenience, we will say that a ring R satisfies "Property D" if every doubly generated ideal in L(R)* is the intersection of powers of prime ideals. We begin with a simple but useful observation. L E M M A 1. Let (R,M) be a quasi-local ring satisfying Property D. If x , yeM then there are only a finite number of primes minimal over (x,y). n Proof (x,y) is the finite intersection of powers of prime ideals, say (x,y) = P) Pf. Then any prime minimal over (x,y) is one of the primes P t . ' = l We also note the following. LEMMA 2. If R satisfies Property D and if S is a multiplicatively closed subset of R, then R s satisfies Property D. Proof. (a/s l ,b/s)R s = (a,b)R s . Glasgow Math. J. 36 (1994) 131-134. https://www.cambridge.org/core/terms. https://doi.
doi:10.1017/s0017089500030640 fatcat:gwtcus2drfggnbhdyosmpuiwqe