Remarks on an exclusive extension generated by a super-primitive element (Q1912847)

Let \(R\) be a Noetherian domain, \(K\) its quotient field, \(L/K\) an algebraic field extension, \(\alpha\in L\), \(d= [K(\alpha):K]\), \(X^d+ \eta_1X^{d-1} +\cdots + \eta_d\) the minimal polynomial of \(\alpha\) over \(K\), \(I_{[\alpha]}= \bigcap^d_{i=1} (R:_R \eta_i)\), \(J_{[\alpha]} =I_{[\alpha]} (1,\eta_1, \dots, \eta_d)\), and \(\widetilde J_{[\alpha]} =I_{[\alpha]} (1,\eta_1, \dots, \eta_{d-1})\). The element \(\alpha\) is said to be super-primitive of degree \(d\) over \(R\) if \(J_{[\alpha]} \nsubseteq p\) for all primes \(p\) of depth one. The authors show that if \(R\) contains an infinite field and if \(\alpha\) is super-primitive over \(R\), then the following statements are equivalent: (i) \(\alpha\) is exclusive over \(R\), i.e., \(R[\alpha] \cap K= R\); (ii) \(\bigcap^{d-1}_{i=1} I_{\eta_i}= I_{\eta_d}\); (iii) grade \(\widetilde J_{[\alpha]}>1\) or \(\widetilde J_{[\alpha]}=R\).