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The distributive hull of a ring

1990-8
Erdoğdu, Vahap
Let R be a commutative ring with identity. An extension MS N of R-modules is said to be distributive if it satisfies the following condition: Mn(X+ Y)=(MnX)+(Mn Y), for all submodules X, Y of N. In [2], Davison has shown that every R-module M which is locally non- zero at every maximal ideal of R has a maximal distributive extension and has raised the question: Is this unique up to M-isomorphism, in which case one can denote it by D(M) and call it the distributive hull of M [l, 51. In this paper we answer the question in the affirmative in the case when M is the R-module R, and we show that D(R) is a ring contained in the maximal quotient ring Q(R) of R such that for each maximal ideal P of R the set of R,-submodules of D(R)p containing R, is linearly ordered. We then describe the distributive hull D(R) in certain cases. In particular, we show that the distributive hull of a Noetherian integrally closed domain R is given by (nPEX RP} n K, where X is the set of all maximal ideals of R of height greater than one and K is the field of quotients of R. If R is an Artinian ring, then D(R) = R. We also show that these results remain true when R is replaced by an ideal (restrictions may be imposed) of R. Throughout R will denote a commutative ring with identity and MaxSpec R will denote the set of maximal ideals of R; if M is a submodule ofan R-module Nandx,yEN, (kf:y)={r~R(ry~M}, (x:y)=(Rx:y). If R is a ring, Z(R) is the set of zero divisors of R. 1 LEMMA~.~. LetRbearingandT={tER-Z(R)IRtERisdistributive}. Then T is a saturated multiplicatively closed subset of R. Proof. Let t, and t2 be any two elements of T. Then clearly t, t, E R - Z(R). Since Rt, G R is distributive, (Rt, : r) + (r : s) = R, for all 263 0021-8693/90 $3.00 Copyright 0 1994 by Academic Press, Inc All rights of reproductmn in any form reserved.