Noetherian Stable Domains

Abstract

A contemporary approach to studying commutative rings is to take a given theorem for abelian groups, then interpret the statement for a ring R. The validity of the new statement for R usually imposes some w x restrictions upon the structure of R. Warfield 32 showed that for torsion-free abelian groups A and B, when A has rank 1, the natural map RFor the remainder of the text, R will denote a noetherian domain. In Section 1 we consider weakly solvable domains and show that R is weakly solvable precisely when each rank 1 module is flat as a module over its Ž endomorphism ring. This in turn is equivalent to R being stable i.e., each . ideal of R is projective over its ring of endomorphisms . In Section 2 we show that stability is equivalent to being solvable as well.Several authors have considered the generalization of Warfield's dualw x wx ity result to noetherian domains 26, 15, 16, 9 . The terminology of 9 states that an integral domain R is a Warfield domain if for any rank 1 module A, the torsion-free modules B of finite rank satisfying 343

Stability in Commutative Rings

“…(ii) Follows immediately from part (i) and [12,Lemma 2].…”

Flat ideals and stability in integral domains

“…(iii) ⇒ (i) follows by applying exactly the same argument used in the proof of [31, Theorem 3.5, (ii) ⇒ (i)]. 2So, in particular, the Noetherian quasi-stable domains are exactly the Noetherian stable domains (cf [19,. Theorem 11]).…”

“…But this case is not interesting since quasi-stable finitely generated ideals are stable (and have already been widely studied especially in the finitely generated case, cf. [19,33]). …”

On the arithmetic of stable domains

“…By Proposition 3.5.4 , orders in quadratic number fields are stable because every ideal is 2-generated (for background on orders in quadratic number fields, we refer to [ 33 ]). Much research was done to characterize domains, for which all ideals are 2-generated ([ 8 , Theorem 7.3], [ 29 , Theorem 17], [ 39 ]). We continue with a characterization within the class of seminormal domains.…”