Consider the ring $R = \displaystyle\frac {\mathbb Z_2[x]}{\langle x^8-1\rangle}$.
i) Is $R$ a finite ring?
ii) Does $R$ have a zero divisor?
iii) Does $R$ have nilpotent elements?
ii) for zero divisor: $[(x^4+1)(x^2+1)(x+1)](x-1) = (x^8-1) = 0 \mod(x^8-1)$. Is this the right way of showing zero divisor?
And I have no idea how we can solve the other two.
Any help would be appreciated.