EUDML

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...

Star operations in extensions of integral domains

David F. Anderson; Said El Baghdadi; Muhammad Zafrullah — 2010

Actes des rencontres du CIRM

An extension $D \subseteq R$ of integral domains is $t$ - (resp., $t$ -) if ${(I R)}^{- 1} = {(I^{- 1} R)}_{v}$ (resp., ${(I R)}_{v} = {(I_{v} R)}_{v})$ for every nonzero finitely generated fractional ideal $I$ of $D$ . We show that strongly $t$ -compatible implies $t$ -compatible and give examples to show that the converse does not hold. We also indicate situations where strong $t$ -compatibility and its variants show up naturally. In addition, we study integral domains $D$ such that $D \subseteq R$ is strongly $t$ -compatible (resp., $t$ -compatible) for every overring $R$ of $D$ .

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A “ $v$ -operation free” approach to Prüfer $v$ -multiplication domains.

Semirigid GCD Domains.

A General Theory of Almost Factoriality.

On Finite Conductor Domains.

On Riesz groups.

The GCD property and irreducible quadratic polynomials.

On some class groups of an integral domain

The Schreier Property and Gauss' Lemma

Star operations in extensions of integral domains