Integral domains with finitely generated groups of divisibility

Abstract

Let D D be an integral domain with integral closure D ¯ \overline D . We show that the group of divisibility G ( D ) G(D) of D D is finitely generated if and only if G ( D ¯ ) G(\overline D ) is finitely generated and D ¯ / [ D : D ¯ ] \overline D /[D:\overline D ] is finite. We also show that G ( D ) G(D) is finitely generated if and only if the monoid of finitely generated fractional ideals of D D (under multiplication) is finitely generated.