Let $\mathbb R$ be the field of real numbers and $P = x \mathbb R [x]$ be the ring of polynomials over $\mathbb R$ with no constant term. Define $R = P \times \mathbb R \times P \times P$ and the addition and multiplication as follows.
$(p, \alpha, p_1, p_2) + (q, \beta, q_1, q_2) = (p + q, \alpha + \beta, p_1 + q_1, p_2 + q_2)$,
$(p, \alpha, p_1, p_2)(q, \beta, q_1, q_2) = (pq + \alpha q + \beta p, \alpha \beta, \alpha q_1 + \beta p_1, \alpha q_2 + \beta p_2)$.
Then $R$ becomes a cummutative ring with identity. Prove the following results:
(1) $N =$ {$a \in R: a$ is nilpotent} is a prime ideal in $R$.
I've already done this!
(2) $R$ contains infinitely many maximal ideals.
This is what I'm having difficulty with. I at first believed $(0, 0, p1, p2)$ to be the maximal ideals of $N$, but then $(p,0,0,0)$ doesn't have a multiplicative inverse in $R/N$. Any advice would be appreciated!
Best Answer
For every $t \in \mathbb{R}$, the map $$\eta_t \colon (p,\alpha,p_1,p_2) \mapsto p(t) + \alpha$$ is a ring homomorphism $R \to \mathbb{R}$. Check that it is surjective, then it follows that $\ker \eta_t$ is a maximal ideal in $R$. Check that for $s\neq t$ we have $\ker \eta_s \neq \ker \eta_t$.