Prime links in Noetherian rings (Q1210496)

It is proved that for a localizable prime ideal \(P\) of a noetherian ring \(R\), either \(P_ P = 0\), or that \(P\) is linked to itself and that its clique consists of \(P\) alone. In the special case when \(R\) is a noetherian prime ring that is a maximal order in its simple artinian quotient ring, any reflexive prime ideal \(P\), being localizable, is thus linked to itself, and the link is via \(P^{(2)}\), the second symbolic power of \(P\) in the sense of \textit{A. Goldie} [J. Algebra 5, 89-105 (1967; Zbl 0154.288)].