Pool problems in ring theory

Consider the additive group of integers ${\bf Z}$.
\begin{enumerate}[label=\alph*)]
	\item Prove that every subgroup of ${\bf Z}$ is a cyclic group.
	\item Prove that every homomorphic image of ${\bf Z}$ is a cyclic group.
	\item Now consider the {\itshape ring} ${\bf Z}$. Exhibit a prime ideal of ${\bf Z}$ that is not maximal.
\end{enumerate}

Let $R$ be an integral domain. Suppose that $a$ and $b$ are non-associate irreducible elements in $R$, and the ideal $(a,b)$ generated by $a$ and $b$ is a proper ideal. Show that $R$ is not a principal ideal domain (PID).

Let $R$ be a commutative ring with identity.  Suppose that for every $a\in R$ there is an integer $n\geq 2$ such that $a^n=a$. Show that every prime ideal of $R$ is maximal.

Let $R$ be a commutative ring with $1$, and $\sigma:R\to R$ be a ring automorphism.
\begin{enumerate}[label=\alph*)]
	\item Show that $F=\{r\in R\mid \sigma(r)=r\}$ is a subring of $R$ (with $1$).
	\item Show that if $\sigma^2$ is the identity map on $R$, then each element of $R$ is the root of a monic polynomial of degree 2 in $F[x]$, where $F$ is as in (a).
\end{enumerate}

Suppose $R$ is a ring such that $r^2=r$ for every element $r\in R$.
\begin{enumerate}[label=\alph*)]
	\item Prove $r=-r$ for every element $r\in R$.
	\item Show $R$ must be commutative. {\itshape Hint:} Consider $(a+b)^2$.
\end{enumerate}

Let $R$ be a commutative ring with $1$. The {\bfseries characteristic} $\operatorname{char}(R)$ of $R$ is the unique integer $n\geq 0$ such that $\langle n\rangle \subset {\bf Z}$ is the kernel of the homomorphism $\theta:{\bf Z}\to R$ given by
\[
	\theta(m)=\begin{cases} \underbrace{1_R+\cdots +1_R}_{m}, & \text{ if }m\geq 0 \\ \underbrace{-1_R+\cdots+-1_R}_{|m|}, & \text{ if }m<0\end{cases}
\]
\begin{enumerate}[label=\alph*)]
	\item Prove that if $f:R\to S$ is a monomorphism of commutative rings with $1$, then $\operatorname{char}(R)=\operatorname{char}(S)$.
	\item Prove by given an example that $\operatorname{char}(R)$ is not always preserved by ring homomorphisms.
\end{enumerate}

\begin{enumerate}[label=\alph*)]
	\item Prove that for every commutative ring with unity, $R$, there is a unique ring homomorphism $\phi_R: {\bf Z}\to R$, and that $\ker(\phi_R)=\langle d_R\rangle$ for some unique nonnegative integer $d_R$. The number $d_R$ is called the {\bfseries characteristic} of $R$ and is denoted $\operatorname{char}(R)$.
	\item Suppose $F_1$ and $F_2$ are fields for which there exists a ring homomorphism $f:F_1\to F_2$. Prove that $\operatorname{char}(F_1)=\operatorname{char}(F_2)$.
\end{enumerate}

Let $A$ be a commutative ring with $1$. The {\bfseries dimension} of $A$ is the maximum length $d$ of a chain of prime ideals $\mathfrak{p}_0\subsetneq \mathfrak{p}_1\subsetneq \cdots \subsetneq \mathfrak{p}_d$. Prove that if $A$ is a PID, the dimension of $A$ is at most 1.

Prove that every Euclidean domain is a principal ideal domain.

Let $F$ be a field and let $\alpha$ be an element that generates a field extension of $F$ of degree five. Prove that $\alpha^2$ generates the same extension.

Let $R$ be a commutative ring with $1$. Use theorems in ring theory to prove:
\begin{enumerate}[label=\alph*)]
	\item $\langle x\rangle$ is a prime ideal in $R[x]$ if and only if $R$ is an integral domain.
	\item $\langle x\rangle$ is a maximal ideal in $R[x]$ if and only if $R$ is a field.
\end{enumerate}

Let $F$ be a field and $F[x]$ be the polynomial ring, which is a principal ideal domain. Let $R=\{f\in F[x]:f'\in (x)\}$, where $(x)\subset F[x]$ is the ideal generated by $x$, and $f'$ is the (formal) derivative of the polynomial $f$. It is a fact that $R$ is a subring of $F[x]$.
\begin{enumerate}[label=\alph*)]
	\item Prove that $x^2$ and $x^3$ are irreducible elements of $R$.
	\item Let $(x^2,x^3)$ be the ideal in $R$ generated by $x^2$ and $x^3$. Prove this is a proper ideal of $R$.
	\item Prove that $(x^2,x^3)$ is not a {\itshape principal} ideal of $R$.
\end{enumerate}

Let $R$ be a commutative ring with 1 and suppose $e\in R$ is {\bfseries idempotent}, i.e., satisfies $e^2=e$.
\begin{enumerate}[label=\alph*)]
	\item Prove that $1-e$ is also idempotent.
	\item Suppose $e\neq 0, 1$. Show that $Re$ and $R(1-e)$ are proper ideals of $R$.
	\item Prove there is an isomorphism $R\cong Re\times R(1-e)$.
\end{enumerate}

An element $r$ of a ring $R$ is said to be {\bfseries idempotent} if $r^2=r$. Suppose that $R$ is a commutative ring with unity containing an idempotent element $e$.

\medskip
\begin{enumerate}[label=(\alph*)]
	\item Prove that $1-e$ is also idempotent.
	\item Prove that $eR$ and $(1-e)R$ are both ideals in $R$ and that
	\[
		R\cong eR\times (1-e)R.
	\]
	\item Prove that if $R$ has a unique maximal ideal, then the only idempotent elements in $R$ are 0 and 1.
\end{enumerate}

Prove that if $\phi:R\to S$ is a surjective ring homomorphism between commutative rings with unity, then $\phi(1_R)=1_S$.

Suppose $R$ is a PID (principal ideal domain). Prove that an ideal $I\subset R$ is maximal if and only if $I=\langle p\rangle$ for a prime $p\in R$. (By definition, an element $p$ is {\bfseries prime} if whenever $p\mid ab$ then $p\mid a$ or $p\mid b$. If you use the fact that prime implies irreducible, you have to prove it.)

Let $R$ be a commutative ring.

\medskip
\begin{enumerate}[label=(\alph*)]
	\item Prove that the set $N$ of all nilpotent elements of $R$ is an ideal.
	\item Prove that $R/N$ is a ring with no nonzero nilpotent elements.
	\item Show that $N$ is contained in every prime ideal of $R$.
\end{enumerate}

Let $R$ be a commutative ring with 1. We say an element $n\in R$ is {\bfseries nilpotent} if there exists a number $k\in {\bf N}$ such that $n^k=0$.
\begin{enumerate}[label=\alph*)]
	\item Show that if $n$ is nilpotent, then $1-n$ is a unit.
	\item Give an example of a commutative ring with 1 that has no nonzero nilpotent elements, but is not an integral domain.
\end{enumerate}

Let $D$ be a principal ideal domain. Prove that every proper nonzero prime ideal is maximal.

Let $I\subseteq {\bf Z}[x]$ denote the set of all polynomials with even constant term.
\begin{enumerate}[label=\alph*)]
	\item Prove that $I$ is an ideal.
	\item Prove that $I$ is not a {\itshape principal} ideal.
\end{enumerate}

Let $R$ be a commutative ring with unity.
\begin{enumerate}[label=\alph*)]
	\item Define what it means for an element in $R$ to be {\bfseries prime}, and also what it means for an element to be {\bfseries irreducible}.
	\item Prove that if $R$ is an integral domain, then every prime element is irreducible.
\end{enumerate}

Let $R$ be a commutative ring with unity, let $I\subseteq R$ be an ideal, and let $\pi:R\to R/I$ be the natural projection homomorphism.
\begin{enumerate}[label=\alph*)]
	\item Show that if $\wp$ is a prime ideal of $R/I$, then $\pi^{-1}(\wp)$ is a prime ideal of $R$.
	\item Show that the assignment $\wp\mapsto\pi^{-1}(\wp)$ is injective on the set of prime ideals of $R/I$.
\end{enumerate}

Let $A$ be a commutative ring with unit. We call $A$ {\bfseries Boolean} if $a^2=a$ for every $a\in A$. Prove that in a Boolean ring $A$ each of the following holds:
\begin{enumerate}[label=(\alph*)]
	\item $2a=0$ for every $a\in A$.
	\item If $I$ is a prime ideal then $A/I$ is a field with two elements (and in particular $I$ is maximal).
	\item If $I=(a,b)$ is the ideal generated by $a$ and $b$ then $I$ can be generated by the single element $a+b+ab$. Conclude that every finitely generated ideal is principal.
\end{enumerate}

Let $R$ be a commutative ring. For each nonempty subset $X\subseteq R$, the {\bfseries annihilator} of $X$ is the set $\operatorname{ann}(X)=\{a\in R\mid ax=0\text{ for all }x\in X\}$.
\begin{enumerate}[label=\alph*)]
	\item Prove that $\operatorname{ann}(X)$ is an ideal of $R$.
	\item Prove that $X\subseteq \operatorname{ann}(\operatorname{ann}(X))$.
\end{enumerate}

Suppose $\phi:R\to S$ is a ring homomorphism, and $S$ has no (nonzero) zero-divisors. Prove from the definitions that $\ker(\phi)$ is a prime ideal.

Let $R$ be a commutative ring with $1$, and $N$ the ideal
\[
	N=\{a\in R\,\mid\, a^n=0\text{ for some }n\}.
\]
Let $[b]$ be the image of $b\in R$ in $R/N$. Prove that if $[a]\in R/N$ and $[a]^m=0$ then $[a]=[0]$.

Let $R$ be a commutative ring. The {\bfseries nilradical} of $R$ is defined to be $N=\{r\in R\,|\, r^n=0\text{ for some }n\in {\bf N}\}$.
\begin{enumerate}[label=(\alph*)]
	\item Prove that $N$ is an ideal of $R$.
	\item Prove that $N$ is contained in the intersection of all prime ideals of $R$.
\end{enumerate}