A look at the prime and semiprime operations of one-dimensional domains. (Q2881001)

Let \(R\) be an integral domain. A \textit{semiprime operation on the ideals of} \(R\) is a closure operation \(c\) on the set of integral ideals of \(R\) (including \(0\)) that satisfies the additional property that \(I^{c}J^{c}\subseteq (IJ)^{c}\) (or equivalently that \( (IJ)^{c}=(IJ^{c})^{c}\) or \((IJ)^{c}=(I^{c}J^{c})^{c}\)). If in addition \(c\) satisfies \((bI)^{c}=bI^{c}\) for all nonzero \(b\) of \(R\), then \(c\) is called a \textit{prime operation} on \(R\). Note that a prime operation necessarily has \(0^{c}=0\) unless \(R\)\ is a field and \(I^{c}=R\) for each ideal \(I\) of \(R\). Similarly one can define a semiprime operation and a prime operation on the set of fractional ideals of \(R\) (including \(0\)). Recall that a star-operation \(\star \) on \(R\) is a closure operation on the set of nonzero fractional ideals of \(R\) that satisfies \(R^{c}=R\), \(I^{c}J^{c}\subseteq (IJ)^{c}\), and \((bI)^{c}=bI^{c}\) for all nonzero \(b\) of \(R\). Clearly \(\star \) becomes\ a prime operation by defining \(0^{\star }=0\). A closure operation on \(R\) is said to be \textit{bounded} if for each maximal ideal \(m\) of \(R\), there is an \(m\)-primary ideal \(I\) such that for all \(m\)-primary ideals \( J\subseteq I\), \(J^{c}=I\). It is shown that for a rank-one discrete valuation domain \(R\), the only semiprime operation on the set of fractional ideals is the identity (but it is shown in a previous paper of the author that there are non-identity semiprime operations on the set of integral ideals of \(R\)) and that for a one-dimensional local Noetherian domain \(R\) there are no bounded semiprime operations on the fractional ideals of \(R\). The author shows that the only prime operation on the set of ideals of the ring \( k[[t^{2},t^{2r+1}]]\), \(k\) a field, is the identity, but that the ring \( k[[t^{3},t^{4},t^{5}]]\) has a prime operation that is not the identity. (Reviewer note: Observe that since the maximal ideal of \(k[[t^{2},t^{2r+1}]]\) is generated by two elements, \(k[[t^{2},t^{2r+1}]]\) is Gorenstein and hence each ideal of \(k[[t^{2},t^{2r+1}]]\)\ is divisorial, that is, \( I=I_{v}=((I)^{-1})^{-1}\). So for any nonzero ideal \(I\) of \( k[[t^{2},t^{2r+1}]]\) we have \(I_{v}=I\subseteq I^{c}\subseteq I_{v}\) (since \( c\) extends to a star-operation also called \(c\)), so \(I=I^{c}\). Hence the only semiprime operation on \(k[[t^{2},t^{2r+1}]]\) (or any one-dimensional Gorenstein domain) is the identity.)