Cancellation properties in ideal systems: A classification of e.a.b. semistar operations
Let \(D\) be a domain with quotient field \(K\) and let \(\star\) be a semistar operation on \(D\). Denote by \(f (D)\) (respectively, \(\overline{F} (D)\)) the set of all finitely generated fractional ideals of \(D\) (respectively, nonzero \(D\)-submodule of \(K\)). A finitely generated fractional ideal \(F\) of \(D\) is called \(\star\)-eab (respectively, \(\star\)-ab) if \((F G)^\star\subset (F H)^\star\) implies that \(G^\star\subset H^\star\), with \(G, H\in f (D)\) (respectively, with \(G, H\in\overline{F} (D)\)) and \(\star\) is called an eab (respectively, ab) semistar operation if each \(F\in f (D)\) is \(\star\)-eab. Let \(\mathfrak w\) be a family of valuation overrings of \(D\). For any \(E\in\overline{F} (D)\), the authors consider the following semistar operations on \(D\): \[ E^{w} =\bigcap \{EW \mid W \in w\}. \] Thus \(w\) is a semistar operation on \(D\). Let \(\star\) be a semistar operation on \(R\). An overring \(V\) of \(D\) is called a \(\star\)-valuation overring of \(D\) if \(F^\star\subset F V\) for all \(F\in f (D)\). The authors define also \(E ^{b(\star)} = \bigcap\{EV \}\), where \(V\) ranges over \(\star\)-valuation overrings of \(R\). Then \(b(\star)\) is also a semistar operation, which is called the complete \(w\)-operation associated with \(\star\). In this paper, the authors prove the implications \(\{\star\) is a complete \(w\)-operation\(\}\Rightarrow \{\star\) is a \(w\)-operation\(\}\Rightarrow \{\star\) is an ab operation\(\}\Rightarrow \{\star\) is an eab operation\(\}\) and the authors point out by giving examples that each of the implications is not reversible.
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