156 Heinzer, Ohm and Pendieton, On integral domains

Chapter 4, p. 165, Ex. 17). Every minimal prime divisor of ^ is a ß^^-prime of A ; and when D is noetherian, the 5^-primes of A coincide with the usual associated primes of A.

5 . 2. Lemma. {B^-primes of proper principal ideals of D} = {minimal prime di- çisors of proper denominator ideals of D}, Moreover^ if Э is a proper denominator ideal of D, then for any y Ф 0 € 9, any minimal prime divisor of 9 is a B^-prime of yD.

Proof . d[xjy) = (y) : X, and any minimal prime divisor of (y) : x h a. ß^-prime of yJD. q. e. d.

If Ж is a set of prime ideals of D and 5 is a multiplicative system in D, we write %g = {QDg\ Q i% and Q r^ S = 0}, In particular, if P is a prime ideal and S = D \P, we write %p = %^ = {QDp | Ç € St and <? < P).

5 . 3. Theorem. Let % be a set of prime ideals not including D itself such that (0) € 2, and as in section 4 let @ (%) = {ideals A of D \ A ^ Q for each Q € Щ, Consider the lowing statements:

( i ) D = n {Dq I Ç € %] (= i)e(^) by 4. 3).

( ii ) Each proper denominator ideal of D is contained in some ideal of î.

( iii ) For each multiplicative system S in D, D^ equals the generalized ring of quotients of Dg with respect to the g. m, s. @(ï^).

( iv ) For each prime ideal P of i), Dp equals the generalized ring of quotients of Dp with respect to the g, m, s, (S(îp).

( v ) Every minimal prime divisor of a proper denominator ideal of D is in %,

( vi ) Every Byfprime of a proper principal ideal of D is in %. Then (i) «=» (ii), and statements (iii) through (vi) are equivalent (and clearly (v) => (ii)). over ^ if D is noetherian and % = {minimal prime ideals of D] ^ {(0)}, then all of the ments are equivalent.

Proof , (i) ^ (ii): For any | € ii: and Ç € ï, I € Dq if and only if 9^(|) ф Q. Therefore D = n {Dq I <? e î} if and only if 9^(|) c: Q for some Q^% whenever è^K\D.

( iii ) ==> (iv) : Trivial.

( iv ) =^ (v): Let P be a minimal prime divisor of 92)(f), f € K\D.

Then PDp is a minimal prime divisor of dp,{^)Dp = 9^> (|). Now use the cation (i) => (ii) with D replaced by Dp and % replaced by %p to conclude that there exists Q' € %p such that 9^^(^) < Q'. But Q' <. PDp, so Q' == PDp. Therefore PDp € %p and hence P ^%.

( v ) ^^ (vi) : Apply 5. 2.

( vi ) =4> (iii): Using the implication (ii)=^(i), it suffices to prove that each proper denominatoi ideal of Dg is contained in some ideal of Xg. But this is immediate from (v), using the fact that 9^)^(1) = dj){i)Dg.

To prove the second assertion of the theorem we apply the following lemma to show that (i) implies (vi) when D is noetherian and % = {minimal primes of D} w {(0)}.

Lemma . Each proper principal ideal of D is a finite intersection of height 1 primary ideals if and only if D = D"^ and each nonzero x^ D is in at most finitely many minimal primes of D.

Proof . See [9], p. 115—116.

5 . 4. Corollary ([5], p. 35, Theorem 47). // % is the set of maximal B^-primes of proper principal ideals of D, then i) == П {Dq \ Q (i%}.