The Schreier Property and Gauss' Lemma

Anderson, Daniel D. ; Zafrullah, Muhammad

Bollettino della Unione matematica italiana, Série 8, 10B (2007) no. 1, pp. 43-62

Résumé

Let $D$ be an integral domain with quotient field $D$. Recall that $D$ is Schreier if $D$ is integrally closed and for all $x, y, z \in D \setminus \{0\}$, $x|yz$ implies that $x = r \cdot s$ where $r|y$ e $s|z$. A GCD domain is Schreier. We show that an integral domain $D$ is a GCD domain if and only if (i) for each pair $a, b \in D \setminus \{0\}$, there is a finitely generated ideal $B$ such that $aD \bigcap bD = B_v$ and (ii) every quadratic in $D[X]$ that is a product of two linear polynomials in $K[X]$ is a product of two linear polynomials in $D[X]$. We also show that $D$ is Schreier if and only if every polynomial in $D[X]$ with a linear factor in $K[X]$ has a linear factor in $D[X]$ and show that $D$ is a Schreier domain with algebraically closed quotient field if and only if every nonconstant polynomial over $D$ is expressible as a product of linear polynomials. We also compare the two most common modes of generalizing GCD domains. One is via properties that imply Gauss' Lemma and the other is via the Schreier property. The Schreier property is not implied by any of the specializations of Gauss' Lemma while all but one of the specializations of Gauss Lemma are implied by the Schreier property.