## Fractional ideals

Last update: 25 June 2012

## Fractional ideals

Let $A$ be a commutative ring. Let $P$ be a finitely generated projective $A-$module. Let $p\in \mathrm{Spec}\left(A\right).$

• The rank of $P$ at $p$, ${\mathrm{rk}}_{p}\left(P\right)$ is the rank of the free ${A}_{p}-$module ${P}_{p}.$
• The rank of $P$ is $n$ if ${\mathrm{rk}}_{p}\left(P\right)=n$ for all $p\in \mathrm{Spec}\left(A\right).$

The set $P(A) = { isomorphism classes cl(P) of finitely generated projective A-modules P of rank 1 }$ with operation $cl(M)+cl(N) = cl( M⊗AN )$ is an abelian group.

Let $A$ be a commutative ring, $S$ a multiplicatively closed subset such that if $s\in S$ then $s$ is not a zero divisor in $A,$ $B={S}^{-1}A.$

An invertible sub $A-$module is a sub $A-$module $M$ of $B$ such that there exists a sub $A-$module $N$ of $B$ such that $M\cdot N=A.$

Let $𝒥 = { invertible A-submodules of B }$ with product multiplication. Then there is an exact sequence of abelian groups ${1} → A× → B× →θ 𝒥 →cl P(A) →φ P(B) a ↦ a b ↦ Ab M ↦ cl(M) ↦ cl(B⊗AM).$

If $A$ is an integral domain, $S=A-\left\{0\right\}$ and $B$ is the field of fractions of $A$ then $𝒥→clP(A) is an isomorphism.$

1. An invertible fractional ideal is an element of $𝒥.$
2. A principal fractional ideal is an element of $\mathrm{im}\theta .$

## Notes and References

The Theorem is given in [Bou, Comm. Ch.II, §5 no.4, Proposition 7] and the definitions and proofs of well-definedness of rank of $P$ are given in [Bou, Comm. Ch.II, §5 no.3].

References?