Let $R$ be a principal ideal domain. Let $a\in R$ be a nonzero, non-unit element. Show that the following are equivalent.
(1) The ideal $(a)$ generated by $a$ is maximal.
(2) The ideal $(a)$ is prime.
(3) The element $a$ is irreducible.
Add to solve later Let $R$ be a finite commutative ring with identity $1$. Prove that every prime ideal of $R$ is a maximal ideal of $R$.
Add to solve later Let $R$ be a commutative ring with $1$.
Suppose that the localization $R_{\mathfrak{p}}$ is a Noetherian ring for every prime ideal $\mathfrak{p}$ of $R$.
Is it true that $A$ is also a Noetherian ring?
Add to solve later Let $R$ be a commutative ring with $1$.
Prove that if every proper ideal of $R$ is a prime ideal, then $R$ is a field.
Add to solve later Let $R$ be an integral domain and let $I$ be an ideal of $R$.
Is the quotient ring $R/I$ an integral domain?
Add to solve later (a) Prove that every prime ideal of a Principal Ideal Domain (PID) is a maximal ideal.
(b) Prove that a quotient ring of a PID by a prime ideal is a PID.
Add to solve later Let $R$ be a commutative ring with identity $1\neq 0$. Suppose that for each element $a\in R$, there exists an integer $n > 1$ depending on $a$.
Then prove that every prime ideal is a maximal ideal.
Add to solve later Give an example of a commutative ring $R$ and a prime ideal $I$ of $R$ that is not a maximal ideal of $R$.
Add to solve later Let
\[P=(2, \sqrt{10})=\{a+b\sqrt{10} \mid a, b \in \Z, 2|a\}\]
be an ideal of the ring
\[\Z[\sqrt{10}]=\{a+b\sqrt{10} \mid a, b \in \Z\}.\]
Then determine the quotient ring $\Z[\sqrt{10}]/P$.
Is $P$ a prime ideal? Is $P$ a maximal ideal?
Add to solve later Let $f: R\to R’$ be a ring homomorphism. Let $P$ be a prime ideal of the ring $R’$.
Prove that the preimage $f^{-1}(P)$ is a prime ideal of $R$.
Add to solve later Let $R$ be a commutative ring with unity.
Then show that every maximal ideal of $R$ is a prime ideal.
Add to solve later Let $R$ be an integral domain and let $S=R[t]$ be the polynomial ring in $t$ over $R$. Let $n$ be a positive integer.
Prove that the polynomial
\[f(x)=x^n-t\]
in the ring $S[x]$ is irreducible in $S[x]$.
Add to solve later Let $R$ be a commutative ring with unity. A proper ideal $I$ of $R$ is called primary if whenever $ab \in I$ for $a, b\in R$, then either $a\in I$ or $b^n\in I$ for some positive integer $n$.
(a) Prove that a prime ideal $P$ of $R$ is primary.
(b) If $P$ is a prime ideal and $a^n\in P$ for some $a\in R$ and a positive integer $n$, then show that $a\in P$.
(c) If $P$ is a prime ideal, prove that $\sqrt{P}=P$.
(d) If $Q$ is a primary ideal, prove that the radical ideal $\sqrt{Q}$ is a prime ideal.
Add to solve later Let $R$ be an integral domain. Then prove that the ideal $(x^3-y^2)$ is a prime ideal in the ring $R[x, y]$.
Add to solve later Let $R$ be a commutative ring with $1$. Show that if $R$ is an integral domain, then the characteristic of $R$ is either $0$ or a prime number $p$.
Add to solve later In the ring
\[\Z[\sqrt{2}]=\{a+\sqrt{2}b \mid a, b \in \Z\},\]
show that $5$ is a prime element but $7$ is not a prime element.
Add to solve later Consider the ring
\[\Z[\sqrt{10}]=\{a+b\sqrt{10} \mid a, b \in \Z\}\]
and its ideal
\[P=(2, \sqrt{10})=\{a+b\sqrt{10} \mid a, b \in \Z, 2|a\}.\]
Show that $p$ is a prime ideal of the ring $\Z[\sqrt{10}]$.
Add to solve later Let $R$ be a commutative ring. Suppose that $P$ is a prime ideal of $R$ containing no nonzero zero divisor. Then show that the ring $R$ is an integral domain.
Add to solve later Let $R$ be the ring of all continuous functions on the interval $[0,1]$.
Let $I$ be the set of functions $f(x)$ in $R$ such that $f(1/2)=f(1/3)=0$.
Show that the set $I$ is an ideal of $R$ but is not a prime ideal.
Add to solve later Let $R$ be a commutative ring with $1$. Prove that the principal ideal $(x)$ generated by the element $x$ in the polynomial ring $R[x]$ is a prime ideal if and only if $R$ is an integral domain.
Prove also that the ideal $(x)$ is a maximal ideal if and only if $R$ is a field.
Add to solve later 
