Duals of ideals in polynomial rings (Q2785944)

Let \(R\) denote a domain, \(K\) its quotient field and \(R[\{X_\alpha\}]\) the polynomial ring in the set \(\{X_\alpha\}\) of indeterminates. Inspite of several investigations on that subject in general it is not known when \(I^{-1}= (I:I)\), \(I\neq 0\) being a (fractional) ideal of \(R\), nor even when \(I^{-1}\) is a ring. Considering the case \(R[\{X_\alpha\}]=:S\) the authors prove among others the following theorem: NEWLINENEWLINENEWLINELet \(I\) be an ideal of \(S\), \(I\neq 0\), for which \(\{x\in R\); \(xy\in I\) for some \(y\in S\setminus I\}=0\). Then the following statements are equivalent NEWLINE\[NEWLINEI^{-1}\text{ is a ring}\iff I^{-1}= (I:I)\iff I^{-1}\subseteq K[\{X_\alpha\}].NEWLINE\]NEWLINE In particular these hold if \(\text{ht }I\geq 2\). For prime ideals \(Q\neq 0\) of \(S\) the problem \(Q^{-1}\) being a ring is answered completely without further assumptions by means of the ideal \(Q\cap R\) of \(R\).NEWLINENEWLINEFor the entire collection see [Zbl 0855.00015].