Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

The following question came up in my research. Since lots of clever people post here, I thought I'd ask it.

Recall that the group ring of a group $G$ is the abelian group $\mathbb{Z}[G]$ consisting of linear combinations of formal symbols $[g]$, where $g$ ranges over elements of $G$ (the abelian group $\mathbb{Z}[G]$ also has an obvious ring structure, but that's not important for this question).

Consider the group ring $\mathbb{Z}[\mathbb{Q}]$ of the rational numbers $\mathbb{Q}$ (considered as an additive group). There is a natural projection $\pi : \mathbb{Z}[\mathbb{Q}] \rightarrow \mathbb{Z}[\mathbb{Q}/\mathbb{Z}]$. It has a large kernel; for instance this kernel contains $[n]-[0]$ for integers $n$ and things like $[3/2]-[1/2]$. There is also a natural involution $i : \mathbb{Z}[\mathbb{Q} \setminus \{0\}] \rightarrow \mathbb{Z}[\mathbb{Q} \setminus \{0\}]$ defined by $i([q]) = [1/q]$. Here by $\mathbb{Z}[\mathbb{Q} \setminus \{0\}]$ I just mean formal sums of $[q]$ where $q$ is a nonzero rational number. We have a natural inclusion $\mathbb{Z}[\mathbb{Q} \setminus \{0\}] \subset \mathbb{Z}[\mathbb{Q}]$.

Question. What is $\text{ker}(\pi) \cap \text{ker}(\pi \circ i)$? It clearly contains things like $[1]-[-1]$, but I don't know if it contains any more "exotic" elements.

share|cite|improve this question
What's $i([0])$? – anon Mar 28 '12 at 3:22
Good point! I'll edit the question to make this make more sense. – Lewis Mar 28 '12 at 3:30
Is this intersection meant to take place in $\mathbb{Z}[\mathbb{Q} \setminus \{ 0 \} ]$? – Qiaochu Yuan Mar 28 '12 at 3:41
@QiaochuYuan : Yes. – Lewis Mar 28 '12 at 3:45

I believe that that intersection of kernels contains, for any integer $k\notin\{0,-1\}$, the element $[1] - [k] + [\frac k{k+1}] - [\frac{-1}{k+1}]$.

I also found (just by messing around) the element $[\frac52] - [\frac57] + [\frac{-2}7] - [\frac23] + [\frac53] - [\frac{-5}2]$.

share|cite|improve this answer
Thanks! This is more complicated than I thought. I'll wait a couple of days to see if anyone gives generators for the kernel and then I'll accept your answer. – Lewis Mar 28 '12 at 14:20

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.