Let $R, S$ be commutative rings with unity . Let $f:R\to S$ be a surjective ring homomorphism
$Q\subseteq S$ be a non-zero pime ideal . Which of the following statements are true?
$(a)f^-(Q)$ is a non-zero prime ideal in $R$
$(b)f^-(Q)$ is a maximal ideal in $R$ if $R$ is a PID
$(c)f^-(Q)$ is a maximal ideal in $R$ if $R$ is a finite commutative ring with unity .
$(d)f^-(Q)$ is a maximal ideal in $R$ if $x^5=x$ for all $x\in R$
$(a)$ True.
Let $r_1r_2 \in f^-(Q)$
Then $f(r_1r_2 )=f(r_1)f(r_2) \in Q$. Since $Q$, is prime ideal, we have either of $ f(r_1),f(r_2)$ belongs to $Q$ .
So either $r_1 \in f^-(Q)$ or $r_2 \in f^-(Q)$
Also $f^-(Q)$ is non-zero and proper since $Q$ is non-zero and $f$ is surjective.
This proves the result.
$(b)$ True since every non-zero prime ideal in a PID is maximal.
$(c)$ True.
$f^-(Q)$ is non-zero prime .
$\Rightarrow R/f^-(Q)$ is finite ID (since $R$ is finite)
and hence a field .
$\Rightarrow f^-(Q)$ is maximal ideal.
$(d)$ I am not sure about this.
Can it be said that order (multiplicative) of every element is $4$ ?
Can it be said that the ring is finite (since order of every element is finite) and then considering the logic in $(c)$ ?
Can it be said that the ring is PID under the given conditions ?
Please help me complete by appropriate hints . Thank you.
Best Answer
Hint for (d): If you have an integral domain $A$ with the property $x^5=x$ for all $x\in A$, then for any $x\ne0$ we have $x^4=1$, so ...
Apply this to $R/f^{-1}(Q)$
I think $R$ might neither be finite nor a PID with the given assumptions, consider e.g. $$R=\Bbb F_5[(x_n)_{n\in\Bbb N}]/\langle (x_n^5-x_n)_{n\in\Bbb N})\rangle$$