# Find a non-zero ideal that is not invertible in $\mathbb{Z}[\sqrt{5}]$.

Find a nonzero ideal that is not invertible in $\mathbb{Z}[\sqrt{5}]$.

I am trying to use I is invertible iff $I(R:I)=R$.

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The definition of $(I:J)$ in Atiyah--MacDonald is that $(I:J)=\{a\in R\mid aJ\subseteq I\}$. Under that definition, $(R:I) = \{a\in R\mid aI\subseteq R\}$, which is all of $R$, and $IR = I$. –  Arturo Magidin Feb 21 '12 at 20:53
@Arturo: Let $K = \mathrm{Frac}(R)$. It is often useful to define $(I:J)$ to be the set of $a \in K$ such that $aJ \subseteq I$. It is with this definition that david35's statement is correct. I would expect any intro book on algebraic number theory would contain this definition. –  David Speyer Feb 21 '12 at 21:16
@david35 What have you learned so far in your course? There are lots of ways to see the answer, including just guessing, but it is difficult to give a hint with no idea what your background is. –  David Speyer Feb 21 '12 at 21:18
@David: I thought it might be a different definition (hence I quoted my own provenance). Thanks. –  Arturo Magidin Feb 21 '12 at 21:22