I am doing some extra practice problems and I got stuck on this one:
Show that every element in the group ring $Z_2 C_n$ with even support (i.e. $wt(u)$ is even) is a zero-divisor. (Hint: Show that $1 + g$ divides $u$).
Why does the hint show this? How might I start?
Hint: look at the ring homomorphism $\varphi$ from $\mathbb{Z}/2\mathbb{Z}[C_n] \rightarrow \mathbb{Z}/2\mathbb{Z}$, defined by $\varphi(\sum_{i=0}^{n-1}a_{i}g^i)=\sum_{i=0}^{n-1}a_{i}$ (where $\langle g\rangle=C_n$). Then the ideal $(1+g) \subseteq ker(\varphi)$. But the reverse inclusion is also true: if $\sum_{i=0}^{n-1}a_{i}=0$, then consider the polynomial $f(X)=\sum_{i=0}^{n-1}a_{i}X^i \in \mathbb{Z}/2\mathbb{Z}[X]$. Since $f(1)=0$, $f$ has a factor $X-1=X+1$. Hence $(1+g)=ker(\varphi)$.