Abstract
Let R be a commutative ring with identity and let P ( R ) be the monoid of principal fractional ideals of R. We show that P ( R ) is finitely generated if and only if P ( R ¯ ) ( R ¯ the integral closure of R) is finitely generated and R ¯ / [ R : R ¯ ] is finite. Moreover, R ¯ is a finite direct product of finite local rings, SPIRs, Bezout domains D with P ( D ) finitely generated, and special Bezout rings S ( S is a Bezout ring with a unique minimal prime P, S P is an SPIR, and P ( S / P ) is finitely generated). Also, P ( R ) is finitely generated if and only if F * ( R ) , the monoid of finitely generated fractional ideals of R, is finitely generated. We show that the monoid F ( R ) of fractional ideals of R is finitely generated if and only if the monoid F ¯ ( R ) of R-submodules of the total quotient ring of R is finitely generated and characterize the rings for which this is the case.
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