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Sia a un intero algebrico con il polinomio minimale $f\left(X\right)$. Si danno condizioni necessarie e sufficienti affinché l'anello $\mathbb{Z}\left[\alpha \right]$ sia seminormale o $t$-chiuso per mezzo di $f\left(X\right)$. Come applicazione, in particolare, si ottiene che se $f\left(X\right)=X^3+aX+b$, $a$, $b\in \mathbb{Z}$ le condizioni sono espresse mediante il discriminante de $f\left(X\right)$.

Let $R$ be a non-maximal order in a finite algebraic number field with integral closure $\overline{R}$. Although $R$ is not a unique factorization domain, we obtain a positive integer $N$ and a family $𝒬$ (called a Cale basis) of primary irreducible elements of $R$ such that $x^N$ has a unique factorization into elements of $𝒬$ for each $x\in R$ coprime with the conductor of $R$. Moreover, this property holds for each nonzero $x\in R$ when the natural map $\mathrm{Spec}\left(\overline{R}\right)\to \mathrm{Spec}\left(R\right)$ is bijective. This last condition is actually equivalent to several properties linked...