I'm working on a question that assumes a group with $|G| >1$ where $G$ is a finite group. I am to show that the ring $\mathbb{Z}_2(G)$ is not a division ring.
I think, more than anything, I am still confused on the notation $\mathbb{Z}_2[G]$. I know I am essentially working with polynomials where the coefficients come from $\mathbb{Z}_2$. I also understand that I'm trying to show that there is a non-zero element that is not invertible, but I am confused on how to use this information in practice.
Thanks in advance!
Pick any element $g\neq e$, say of order $n$. Consider the element $$x=\sum_{i=1}^n{1g^i}$$ What is $1g\cdot x$? We compute $$1g\cdot x=\sum_{i=1}^n{1g^{i+1}}=x$$ Since $1g\cdot x = x$, we have that $$(1g-1e)\cdot x = x-x = 0$$ Thus the ring has zero divisors.