In the context of algebraic number theory, a **fractional ideal** is a generalization of the notion of an ideal in a ring. Specifically, fractional ideals are particularly useful in the study of Dedekind domains and more generally in the structure of arithmetic in number fields. ### Definitions and Properties 1. **Integral Domain**: First, consider a domain \( R \), typically a Dedekind domain, which is an integral domain where every nonzero proper prime ideal is maximal.
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For an integral domain with fraction field , a fractional ideal is a nonzero -submodule for which for some . A finitely generated submodule of has a common denominator and is therefore a fractional ideal. The product is generated by pairwise products of elements. An invertible fractional ideal has a multiplicative inverse and is finitely generated; arbitrary fractional ideals need not be finitely generated over a non-Noetherian domain. The familiar fractional ideals of a number field are the special case . This general definition also applies to the ideals representing Cartier divisors on affine integral schemes.