Normality, seminormality and quasinormality of \({\mathbb{Z}}[^ n\sqrt{m}]\) (Q581459)

Recall that if R is a reduced Noetherian ring such that the canonical homomorphism \(Pic(R)\to Pic(R[X])\) (resp., \(Pic(R)\to Pic(R[X\), \(X^{- 1}]))\) is an isomorphism, then R is called seminormal (resp., quasinormal). If the kernel of \(Pic(R)\to Pic(R[X])\) has no p-torsion for p an integer, R is called p-seminormal. Let m and \(n\geq 2\) be integers, and let \(\alpha\) be an n-th root of m. This paper studies when \({\mathbb{Z}}[\alpha]\) is normal, quasinormal, seminormal, or p-seminormal. For example, it is shown that \({\mathbb{Z}}[\alpha]\) is normal if and only if m is square-free, and for all prime divisors p of n, \(p^ 2\) does not divide \(m^ p-m\). Also, \({\mathbb{Z}}[\alpha]\) is seminormal if and only if one of the following two statements holds: (a) m is square-free, and for each prime divisor p of n, \(p^ 2\) does not divide both n and \(m^ p- m\). (b) \(n=2,\) \(m=ab^ 2\) where a and \(b>1\) are square-free, relatively prime integers, with b odd.