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What exactly are the right ideals of $L=H(\mathbb Z)=\{\,a+bi+cj+dk;\quad a,b,c,d \in \mathbb Z\}$ (the Lipschitz quaternions)?

we can see here

Ideal class "group" of Lipschitz (fully-integer) quaternions

These are $(\alpha)$ , $(\alpha, \alpha\frac{1+i+j+k}{2})$. I am confused...Is it correct? I can't understand its proof!! How the last is possible? while $\frac{1}{2}\not\in\mathbb Z$!!

IS $(\alpha, \alpha\frac{1+i+j+k}{2})$is a subset of $L$ at all?

leila
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  • $\frac12(1+i+j+k)\alpha$ might lie in $L$. Try $\alpha=1+i$. – Angina Seng May 04 '19 at 09:56
  • Can somebody please recommend me a more official reference such as a paper or a book for the explain of right ideals of the Lipschitz quaternions? I can find it just in this page. Or more explain for the proof in this page?

    https://math.stackexchange.com/questions/1964945/ideal-class-group-of-lipschitz-fully-integer-quaternions

     – leila May 24 '19 at 17:16

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