# fractional ideals

If $D$ is a domain and $K$ is field then for $x\in K$, $xD$ is a fractional ideals of $D$. If $xD$ and $yD$ are two fractional ideals then is true or not $xyD\subseteq xD$ ?. Thanks

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What about $D = \mathbf Z$, $x = 2$, $y = 1/2$? –  Dylan Moreland Mar 29 '12 at 20:18
Dylan's example pretty much wraps up the question, but we can come up with another point of view: suppose $$xyD\subset xD\Longrightarrow \forall d\in D\,\,\exists\,d'\in D\,\,\,s.t.\,\,\,xyd=xd'\Longrightarrow x(yd-d')=0\Longrightarrow$$ $$\Longrightarrow x=0\,\,\,or\,\,\,yd=d'$$as we're in a domain or in its fractions field.
If $\,x=0\,$ the claim is trivially true (as $\,0\cdot yD=\{0\}\subset 0\cdot D\,$), otherwise $$\forall d\in D\,\,\,\exists\,d'\in D\,\,\,s.t.\,\,\,\,yd=d'$$and taking $\,d=1\in D\,$ we get that $\,y\in D\,$, which of course could be not the case.