$R$ be a commutative ring with unity , $|R^{\times}|=p$ an odd prime then is $R$ isomorphic to a quotient of $\mathbb Z_2[x]/\langle x^p-1\rangle$?

Question

Let $R$ be a commutative ring with unity such that its group of invertible elements has order $p$ an odd prime , then is it true that there exist a surjective ring homomorphism from $\mathbb Z_2[x]/\langle x^p-1\rangle$ onto $R$ ? I can see that since $-1\in R^{\times}$ and $(-1)^2=1$ and $2$ does not divide $p$ , so $-1=1$ , but I cannot proceed further . Please help . Thanks in advance .

Answer 1

The answer is No. The unit group of $\mathbb F_4[x]$ has size $3$, but this infinite ring is not a quotient of the $8$-element ring $\mathbb F_2[x]/(x^3-1)$.

But if $R$ is a commutative unital ring with $|R^{\times}|=p$, $p$ odd, then $R$ must be of characteristic $2$ (as you observed). If $R^{\times}$ is generated by $u$, then there is a homomorphism $\mathbb F_2[x]\to R\colon x\mapsto u$ which has kernel containing $x^p-1$. Thus there is a homomorphism $\mathbb F_2[x]/(x^p-1)\to R$ whose image contains $R^{\times}$. But, as the example shows, the image need not be all of $R$.