Pool problems in ring theory
Consider the additive group of integers
- Prove that every subgroup of
is a cyclic group. - Prove that every homomorphic image of
is a cyclic group. - Now consider the ring
. Exhibit a prime ideal of that is not maximal.
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
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
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
- Show that
is a subring of (with ). - Show that if
is the identity map on , then each element of is the root of a monic polynomial of degree 2 in , where is as in (a).
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}
A Boolean algebra is a ring
for all .- Every prime ideal
is maximal, and is a field with two elements.
A {\bfseries Boolean algebra} is a ring $A$ with $1$ satisfying $x^2=x$ for all $x\in A$. Prove that in a Boolean algebra $A$:
\begin{enumerate}[label=\alph*)]
\item $2x=0$ for all $x\in A$.
\item Every prime ideal $\mathfrak{p}$ is maximal, and $A/\mathfrak{p}$ is a field with two elements.
\end{enumerate}
Suppose
- Prove
for every element . - Show
must be commutative. Hint: Consider .
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
- Prove that if
is a monomorphism of commutative rings with , then . - Prove by given an example that
is not always preserved by ring homomorphisms.
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}
- Prove that for every commutative ring with unity,
, there is a unique ring homomorphism , and that for some unique nonnegative integer . The number is called the characteristic of and is denoted . - Suppose
and are fields for which there exists a ring homomorphism . Prove that .
\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
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.
Prove that every Euclidean domain is a principal ideal domain.
Suppose
Suppose $R$ is a finite ring with no nontrivial zero-divisors. Prove that $R$ contains an element $1$ satisfying $1\cdot a=a\cdot 1=a$ for all $a\in R$.
Let
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
is a prime ideal in if and only if is an integral domain. is a maximal ideal in if and only if is a field.
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
- Prove that
and are irreducible elements of . - Let
be the ideal in generated by and . Prove this is a proper ideal of . - Prove that
is not a principal ideal of .
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
- Prove that
is also idempotent. - Suppose
. Show that and are proper ideals of . - Prove there is an isomorphism
.
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
-
Prove that
is also idempotent. -
Prove that
and are both ideals in and that -
Prove that if
has a unique maximal ideal, then the only idempotent elements in are 0 and 1.
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
Prove that if $\phi:R\to S$ is a surjective ring homomorphism between commutative rings with unity, then $\phi(1_R)=1_S$.
Suppose
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
- Prove that the set
of all nilpotent elements of is an ideal. - Prove that
is a ring with no nonzero nilpotent elements. - Show that
is contained in every prime ideal of .
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
- Show that if
is nilpotent, then is a unit. - Give an example of a commutative ring with 1 that has no nonzero nilpotent elements, but is not an integral domain.
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
Let $D$ be a principal ideal domain. Prove that every proper nonzero prime ideal is maximal.
Let
- Prove that
is an ideal. - Prove that
is not a principal ideal.
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
- Define what it means for an element in
to be prime, and also what it means for an element to be irreducible. - Prove that if
is an integral domain, then every prime element is irreducible.
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
- Show that if
is a prime ideal of , then is a prime ideal of . - Show that the assignment
is injective on the set of prime ideals of .
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
for every .- If
is a prime ideal then is a field with two elements (and in particular is maximal). - If
is the ideal generated by and then can be generated by the single element . Conclude that every finitely generated ideal is principal.
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
- Prove that
is an ideal of . - Prove that
.
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
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
Let
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
- Let
be ideals, and put . Use the First Isomorphism Theorem to prove that . - Prove the prime ideals of
have the form where is a prime ideal for . (Omit the proof that this is an ideal.)
Let $R_1,\ldots, R_k$ be commutative rings, and set $R=R_1\times \cdots \times R_k$.
\begin{enumerate}[label=\alph*)]
\item Let $I_j\subset R_j$ be ideals, and put $I=I_1\times \cdots \times I_k$. Use the First Isomorphism Theorem to prove that $R/I\simeq R_1/I_1\times \cdots \times R_k/I_k$.
\item Prove the prime ideals of $R$ have the form $R_1\times \cdots \times R_{j-1}\times P_j\times R_{j+1}\times \cdots \times R_k$ where $P_j\subset R_j$ is a prime ideal for $1\leq j\leq k$. (Omit the proof that this is an ideal.)
\end{enumerate}
Let
- Prove that
is an ideal of . - Prove that
is contained in the intersection of all prime ideals of .
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}
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