Let $M=\mathbf{Z}[C_n]$ have a $\mathbf{Z}$-basis $\{e^1,\ldots,e^{2n-1}\}$, then I want to show the following map is $\mathbf{Z}$-bilinear. $\varphi:M\times M\to\mathbf{Z}$
$(e^r,e^s)\mapsto\begin{cases}1,&r={s+1}\\-1,&s=r+1\\0,&o/w\end{cases}\\$
Does anyone have any thoughts? I was thinking of breaking this up into two maps as follows:
$(e^r,e^s)\mapsto e^re^{-s}\mapsto\begin{cases}sgn(r-s)\\0\end{cases}$
The problem then arises about how to show linearity in the first variable. I get,
$(e^r+e^s,e^k)\mapsto e^{r-k}+e^{s-k}\mapsto ???$
What do I do next?