I am trying to understand this paper of D. I. Wallace on conjugation of integer matrices by $SL_n \Bbb Z$, cited in these notes by K. Conrad about the Latimer-MacDuffee theorem.
Let $I$ and $J$ be two fractional ideals of an order $O$ in a number field $K$. In both references, $I$ and $J$ are said to be narrowly equivalent if $J = rI$ for some element $r \in K$ of positive norm. The equivalence classes arising from this relation are denominated narrow-ideal classes and the amount of such classes in defined as the narrow-class number.
On the other hand, the narrow class group $Cl^+(O)$ is defined by the relation $I \sim J$ if $J = r I$ for a totally positive element $r$, i.e., an element such that $\sigma(r) > 0$ for each real embedding $\sigma \colon K \to \Bbb R$. The equivalence classes of this relation are also called narrow-ideal classes.
Are these two definition of narrowly equivalent the same? That is, given an element of positive norm $r$, is there a totally positive element $s$ such that $r I = s I$?
The answer is no. Take a totally real cubic number field; then $rI = sI$ is equivalent with $(r) = (s)$, i.e., $r$ and $s$ differ by a unit. This is impossible if the unit group modulo $-1$ consists of totally positive units.