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The Schreier Property and Gauss' Lemma

Daniel D. AndersonMuhammad Zafrullah — 2007

Bollettino dell'Unione Matematica Italiana

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 D { 0 } , x | y z implies that x = r 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 D { 0 } , there is a finitely generated ideal B such that a D b D = 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. AndersonSaid El BaghdadiMuhammad Zafrullah — 2010

Actes des rencontres du CIRM

An extension D 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 R is strongly t -compatible (resp., t -compatible) for every overring R of D .

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