Fractional ideal

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Given an integral domain $R$ with field of fractions $K$, a fractional ideal, $I$, of $R$ is an $R$-submodule of $K$ such that $\alpha I\subseteq R$ for some nonzero $\alpha\in R$. Explicitly, $I$ is a subset of $K$ such that for any $x,y\in I$ and $r\in R$:

  • $x+y\in I$
  • $rx \in I$
  • $\alpha x\in R$

To prevent confusion, regular ideals of $R$ are sometimes called integral ideals. Clearly any integral ideal of $R$ is a fractional ideal of $R$ (simply take $\alpha = 1$). Moreover, it is easy to see that a fractional ideal, $I$, of $R$ is an integral ideal of $R$ iff $I\subseteq R$.

Addition and multiplication of fractional ideals can now be defined just as they are for integral ideals. Namely, for fractional ideals $I$ and $J$ we let $I+J$ be the submodule of $K$ generated by the set $I\cup J$ and let $IJ$ be the submodule of $K$ generated by the set $\{xy|x\in I, y\in J\}$. Note that if $\alpha I,\beta J\subseteq R$ then $\alpha\beta(I+J),\alpha\beta(IJ)\subseteq R$, so these are indeed fractional ideas.

Fractional ideals are of great importance in algebraic number theory, specifically in the study of Dedekind domains.

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