Stability, duality, 2-generated ideals and a canonical decomposition of modules
In this paper the author studies Warfield (and totally divisorial) domains with a notion that is more general than total divisoriality, that of stability. A non-zero ideal of an integral domain is stable if it is projective over its ring of endomorphisms. If every non-zero ideal of \(R\) is stable, then \(R\) is said to be stable. It is well-known that if every ideal of a Macaulay ring can be generated by two elements, then \(R\) is stable [\textit{H. Bass}, Math. Z. 82, 8--28 (1963; Zbl 0112.26604)]. The converse fails, and consequently, even for Noetherian integral domains, stability is weaker than the property that every ideal be 2-generated. The author frames the concept of divisoriality in terms of stability. Since every totally divisorial domain is stable but not vice versa, the class of totally divisorial domains is properly situated between the classes of stable and Warfield domains. In the present paper, the author uses this observation to bridge the two classes of integral domains.
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