Star operations and pullbacks. (Q1429913)

This paper considers star operations on pullbacks. Recall that a star operation on an integral domain \(T\) is a closure operation \(\star \) on the set of nonzero fractional ideals of \(T\) that satisfies \(T^{\star }=T\) and \((xA)^{\star }=xA^{\star }\) for all nonzero elements of the quotient field of \(T\) and nonzero fractional ideals \(A\) of \(T\). Let \(T\)\ be an integral domain, \(M\) an ideal of \(T\), \(k=T/M\), \(D\) a subdomain of \(k\), \( \varphi :T\rightarrow k\) the natural map, and \(R\) the pullback \(\varphi ^{-1}(D)\). Given star operations \(\star _{D}\) on \(D\) and \(\star _{T}\) on \(T\) , a natural star operation \(\diamond \) is defined on \(R\) which is the meet of a ``lifting'' of \(\star _{T}\)\ and a ``projection'' of \(\star _{D}\). In the case where \(M\) is a maximal ideal of \(T\), i.e., the \(D+M\) construction, it is shown that the projection map from the set of star operations on \(D+M\)\ to those on \(D\) is surjective, thus answering a question raised by \textit{D. F. Anderson} [Rend. Circ. Mat. Palermo, II. Ser. 41, 221--230 (1992; Zbl 0777.13004)]. This paper contains many interesting examples and should be read by anyone interested in star operations.