Free groups of ideals

Abstract We study the freeness of the group $$\textrm{Inv}(D)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>Inv</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>D</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> of invertible ideals of an integral domain D , and the freeness of some related groups of (fractional) ideals. We study the relation between $$\textrm{Inv}(D)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>Inv</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>D</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and $$\textrm{Inv}(D_P)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>Inv</mml:mtext> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>D</mml:mi> <mml:mi>P</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , in particular in the locally finite case, and we analyze in more detail the case where D is Noetherian (obtaining a characterization of when $$\textrm{Inv}(D)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>Inv</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>D</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is free for one-dimensional analytically unramified Noetherian domains) and where D is Prüfer.