Throughout this page, all rings are commutative with $1 \neq 0$. You may assume that for any ring $R$, the set of nilpotent elements in $R$ equals the intersection of all prime ideals of $R$.
Ring theory
1. Let $k$ be a field.
i) Does $k[[x]][y]=k[y][[x]]$ ?
ii) Does $k((x))((y))=k((x, y))$ ?
2. Let $R$ be a Noetherian local ring.
i) If $r \in R$ is a nonunit, show that $\bigcap_{n \geq 1}\left(r^{n}\right)=0$.
ii) If $r \in R$ is a nonzerodivisor such that $R /(r)$ has no nilpotents, show that $R$ has no nilpotents.
3. Let $R$ be a ring, and $R[x]$ the polynomial ring over $R$ in 1 variable.
i) If $f \in R[x]$ is a zerodivisor, show that there exists $0 \neq a \in R$ with $a f=0$.
ii) Show that every maximal ideal in $R[x]$ contains a nonzerodivisor.
iii) Show that (nil $R$ ) $[x]=$ nil $R[x]=\operatorname{Rad} R[x]$, and deduce that $R[x]$ has infinitely many maximal ideals (here nil denotes the nilradical = set of nilpotents).
4. Let $R$ be a ring. If every prime ideal of $R$ is principal, show that every ideal is principal (in this case, $R$ is a PID iff $R$ has no nilpotents or idempotents).
5. For a ring $R$, let $\operatorname{Min}(R)$ be the set of minimal primes of $R$, i.e. prime ideals that are minimal with respect to inclusion. Notice that by Zorn's lemma, $\operatorname{Min}(R) \neq \emptyset$.
i) Show that $\bigcup_{p \in \operatorname{Min}(R)} p \subseteq\{$ zerodivisors in $R\}$.
ii) Show that equality holds in (i) if $R$ has no nilpotents.
Past exams
6. (6.12.26) Is there a ring that has exactly 5 units?
7. i) (2012) Let $R$ be a ring. Show that $R$ is a field iff there is a monic polynomial $f \in R[x]$ such that $(f)$ is a maximal ideal in $R[x]$.
ii) Show by example that i) need not hold if $f$ is not required to be monic.
8. (6.11.17, 6.9.3) Find all ring automorphisms of (i) $\mathbb{Z}[x]$, and (ii) $\mathbb{R}$.
9. (6.10.7) Let $R$ be a ring, $a \in R, n, m \in \mathbb{N}$, and $d:=\operatorname{gcd}(n, m)$. Show that $\left(a^{n}-1, a^{m}-1\right)=$ $\left(a^{d}-1\right)$.
10. (6.9.13) Show that there is no ring with additive group isomorphic to $\mathbb{Q} / \mathbb{Z}$.
Ring theory
1. Let $k$ be a field.
i) Does $k[[x]][y]=k[y][[x]]$ ?
ii) Does $k((x))((y))=k((x, y))$ ?
2. Let $R$ be a Noetherian local ring.
i) If $r \in R$ is a nonunit, show that $\bigcap_{n \geq 1}\left(r^{n}\right)=0$.
ii) If $r \in R$ is a nonzerodivisor such that $R /(r)$ has no nilpotents, show that $R$ has no nilpotents.
3. Let $R$ be a ring, and $R[x]$ the polynomial ring over $R$ in 1 variable.
i) If $f \in R[x]$ is a zerodivisor, show that there exists $0 \neq a \in R$ with $a f=0$.
ii) Show that every maximal ideal in $R[x]$ contains a nonzerodivisor.
iii) Show that (nil $R$ ) $[x]=$ nil $R[x]=\operatorname{Rad} R[x]$, and deduce that $R[x]$ has infinitely many maximal ideals (here nil denotes the nilradical = set of nilpotents).
4. Let $R$ be a ring. If every prime ideal of $R$ is principal, show that every ideal is principal (in this case, $R$ is a PID iff $R$ has no nilpotents or idempotents).
5. For a ring $R$, let $\operatorname{Min}(R)$ be the set of minimal primes of $R$, i.e. prime ideals that are minimal with respect to inclusion. Notice that by Zorn's lemma, $\operatorname{Min}(R) \neq \emptyset$.
i) Show that $\bigcup_{p \in \operatorname{Min}(R)} p \subseteq\{$ zerodivisors in $R\}$.
ii) Show that equality holds in (i) if $R$ has no nilpotents.
Past exams
6. (6.12.26) Is there a ring that has exactly 5 units?
7. i) (2012) Let $R$ be a ring. Show that $R$ is a field iff there is a monic polynomial $f \in R[x]$ such that $(f)$ is a maximal ideal in $R[x]$.
ii) Show by example that i) need not hold if $f$ is not required to be monic.
8. (6.11.17, 6.9.3) Find all ring automorphisms of (i) $\mathbb{Z}[x]$, and (ii) $\mathbb{R}$.
9. (6.10.7) Let $R$ be a ring, $a \in R, n, m \in \mathbb{N}$, and $d:=\operatorname{gcd}(n, m)$. Show that $\left(a^{n}-1, a^{m}-1\right)=$ $\left(a^{d}-1\right)$.
10. (6.9.13) Show that there is no ring with additive group isomorphic to $\mathbb{Q} / \mathbb{Z}$.