Putting \(t\)-invertibility to use (Q2752919)

Let \(D\) be an integral domain with quotient field \(K\) and let \(F(D)\) be the set of non-zero fractional ideals of \(D\). A star-operation * on \(D\) is a closure operation on \(F(D)\) that satisfies \(D^*=D\) and \(xA^*= (xA)^*\) for all nonzero \(x\in K\) and \(A\in F(D)\). Two important star-operations are the \(v\)-operation \(I_v=(I^{-1})^{-1}\) where as usual \(I^{-1}= \{x\in K\mid xI \subseteq D\}\) and its finitary companion \(I_t=\bigcup \{J_v\mid J\in F(D)\) is finitely generated and \(J\subseteq I\}\). A fractional ideal \(I\) is said to be *-invertible if \((II^{-1})^*=D\). Thus \(I\) is \(t\)-invertible if \((II^{-1})_t=D\). The set \(\text{Inv}_t(D)\) of \(t\)-invertible \(t\)-ideals forms a group under the \(t\)-product \(I*J= (IJ)_t\). Finally, the \(t\)-class group of \(D\) is the quotient group Cl\(_t(D)=\text{Inv}_t(D)/P(D)\) where \(P(D)\) is the subgroup of non-zero principal fractional ideals of \(D\).NEWLINENEWLINENEWLINEThis paper begins with an introduction to star operations in general and to the \(v\)- and \(t\)-operations in particular. Then *-, \(v\)-, and \(t\)-invertible ideals are surveyed. Other topics covered include the \(w\)-operation \((I_w=\{x\in K\mid xJ \subseteq I\) for some ideal \(J\) where \(J_t=D\})\), the \(t\)-class group, PVMD's, integral domains and star operations defined by locally finite intersections of localizations, and many more.NEWLINENEWLINENEWLINEThis lively article written by one of the leading experts on the subject is recommended reading for anyone who wants to know (more) about the \(t\)-operation and its uses. A fairly complete list of references numbering almost 100 articles is given.NEWLINENEWLINEFor the entire collection see [Zbl 0964.00012].