# Ring of Lipschitz Integers

Let $L$ be the ring of Lipschitz Integers and $a, b, c, d\in L$. Considere $L$ as a left $L$-module and let $(a), (b), (c), (d)$ be the left submodules generated by $a, b, c,$ and $d$, respectively. It is true that the factor modules

$$\frac{L}{(c)+(d)}\;\;and\;\;\frac{(a)+(b)}{(ca)+(db)},$$

are isomorphic?

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For those who don't know: a Lipschitz integer is a quaternion with integer coefficients. – André Henriques Jun 10 '12 at 20:23
Are b and c allowed to be units? – Zack Wolske Jun 10 '12 at 20:43
It is easy to prove that $\frac{L}{(a)}\cong\frac{(c)}{(ac)}$, for $a, c\neq 0$, and the question is a generalization of this fact. If $a$ or $c$ are units this fact is trivial. So, assume, in the question, that $a, b, c, d$ are not units. – zacarias Jun 10 '12 at 21:08
If both $b$ and $c$ are units, then you're asking if $L/L$ and $L/(a) + (d)$ are isomorphic. Or if you don't want to use units, take $a=d$, $b=c$, and $a, b$ coprime. – Zack Wolske Jun 10 '12 at 21:16
Thank you Zack for your comments. The condiction $a$ and $b$ coprime is not well-defined in $L$ since $L$ is not a principal domain. But now I think the answer to the question is NO. Now I'm trying to prove that $$\frac{L}{(\bar{a})+(\bar{b})}\cong\frac{ (a) +(b)}{(\bar{a}a)+(\bar{b}b)}$$ – zacarias Jun 10 '12 at 21:56