---
title: "Commutative Algebra Problems"
sort: 020
---

Tags: #study-guides

# Problems

## Undergrad

 - [ ] Show that prime implies irreducible, and the converse only holds for UFDs.
 - [ ] Show that maximal implies prime but not conversely, so $\mspec R \subseteq \spec R$.
 - [ ] Give an example of a non-principal ideal.
 - [ ] Show that if $R$ is a UFD then $R[x]$ is a UFD.
 - [ ] Prove Gauss' lemma.
 - [ ] Show that if a ring $R$ has factorization into irreducibles and irreducibles are prime, then $R$ is a UFD.
 - [ ] Show that $\kxn$ is a UFD.
 - [ ] Show that nonzero proper principal ideals in $\kxn$ are generated by irreducible polynomials.
 - [ ] Describe $\spec A$ for $A \da R\plocalize{\mfp}$, and show that $A$ is local with $\mfm = \mfp_e$.
 - [ ] Show that $M\tensor_R N$ may equal zero when $M\neq 0, N\neq 0$.
 - [ ] Find a polynomial $f\in \CC[x]$ such that $f(\ZZ) \subseteq \ZZ$ but $f \not\in \ZZ[x]$.
 - [ ] Show that primes are principal in a UFD.
 - [ ] Show that UFD and PID are equivalent for Dedekind domains.
 - [ ] Show that PID implies UFD.
 - [ ] Give a necessary and sufficient condition for an irreducible polynomial to be [inseparable](inseparable).
 - [ ] Show that in characteristic zero, algebraic implies [separable](separable.md) for field extensions.
 - [ ] Show that a finitely generated torsionfree $R\dash$module need not be free if $R$ is not a PID.
 - [ ] Show that the dual of a finitely generated module need not be finitely generated.
 - [ ] Give examples of non-[Noetherian](Noetherian.md) rings.
 - [ ] Show that $I\in \Id(R)$ is prime iff $R\sm I$ is a submonoid of the multiplicative monoid of $R$, i.e. $R\sm I$ is multiplicatively closed and contains $1_R$.

## Ring Basics

 - [ ] Show that every [integral domain](integral%20domain) is [reduced](reduced.md).
 - [ ] Give an example of a ring that is not [reduced](reduced.md).
 - [ ] Show that $I$ is a [radical ideal](radical%20ideal) iff $A/I$ is [reduced](reduced.md).
 - [ ] Show that $R$ Noetherian implies $R[[x]]$ [Noetherian](Noetherian.md).
 - [ ] Show that $\sqrt{I}$ is an ideal if $I$ is an ideal.
 - [ ] Find a ring satisfying the ACC but not the DCC, and vice-versa.
	 - [ ] Find a ring that satisfies neither the ACC nor the DCC.
 - [ ] Show that the [ideal correspondence](ideal%20correspondence) preserves $\spec, \mspec,$ and radical ideals.
 - [ ] Show that if $a, b$ are [radical ideals](radical%20ideals) then $a\intersect b$ is radical but $a+b$ need not be.
 - [ ] Show that for a [local](local) ring $(R, \mfm)$, $\jacobsonrad{R} = \mfm$.
 - [ ] Show that in a UFD, prime ideals of [height](height.md) 1 are principal.
 - [ ] Show that [regular rings](regular%20rings) are [Cohen-Macaulay](Cohen-Macaulay.md).
 - [ ] (Standard exercise) Let $K$ be a field. A commutative $K$-algebra of finite dimension is semisimple if and only if it is [reduced](reduced.md).

## Ideals

 - [ ] Show that $R$ is a UFD iff every [height](height.md) 1 prime ideal is principal.
 - [ ] Show that powers of a maximal ideal $\mfm$ are $\mfm\dash$[primary](primary.md).
 - [ ] Show that if $A\leq B$ is a subring and $\mfp \in \spec A$ then there exists $\mfq \in \spec B$ such that $\mfq \intersect A = \mfp$.
 - [ ] Show that an ideal $I\normal \kxn$ is homogeneous iff it is graded, i.e. $I = \bigoplus I_d$ where $I_d \da I \intersect \kxn_d$, the homogeneous degree $d$ part of the graded ring $\kxn$.
 - [ ] Let $k[V]$ be the coordinate ring of a variety, and show that every maximal ideal $\mfm\in \mspec k[V]$ is of the form $\mfm_p \da \ts{f\in k[V] \st f(p) = 0}$ for some point $p\in V$.
 - [ ] Show that every $I\in \Id(R)$ is a projective $R\dash$module.


## Local Rings

 - [ ] Given $(A, \mfm_A)$, show that $A\localize{\mfm}$ is [local](local). What is its maximal ideal?
 - [ ] Show that for a fixed $S \subseteq R$, there is an exact functor $S\inv: \mods{R} \to \mods{S\inv R}$.
 - [ ] Show that the following are local properties:
	 - [ ] Being zero, i.e. $M= 0$ if $M\localize{\mfm} = 0$ for all $\mfm\in \mspec R$ for $\rmod$.
	 - [ ] Injectivity of module morphisms, i.e. $M\to N$ is injective iff $M\localize{\mfm} \to N\localize{\mfm}$ is injective for all $\mfm \in \mspec R$.
	 - [ ] Being [reduced](reduced.md): $R$ is reduced iff $R\localize{\mfm}$ is reduced for all $\mfm\in \mspec R$.
	 - [ ] [Flatness](Flatness)
	 - [ ] Exactness, i.e. $A\to B\to C$ is exact iff $A\localize{\mfm} \to B \localize{\mfm} \to C\localize{\mfm}$ is exact for all $\mfm \in \mspec R$
	 - [ ] Being [integrally closed](integrally%20closed.md).
	 - [ ] Being coprime ideals, i.e. $I+J = R \iff I_\mfp + J_\mfp = R_\mfp$ for all $\mfp\in \spec R$.
 - [ ] When is a [Unsorted/localization of rings](Unsorted/localization%20of%20rings.md) a subring of the [fraction field](fraction%20field)?
 - [ ] Show that if $A \rightarrow B$ is a ring homomorphism and $M$ is a flat $A$-module, then $M_{B}=B \otimes_{A} M$ is a flat B-module. 
	(Use the canonical isomorphisms (2.14), (2.15).)
 - [ ] Show that $(A/I)\localize{S} \cong M/IM$ for $M\da A\localize{S}$

## Noetherian Rings

![Exercises in Noetherian rings](Unsorted/Noetherian.md#Exercises)


## Modules and Algebras

 - [ ] Show that a morphism $A\to B$ is the same as giving $B$ an $A\dash$algebra structure.
 - [ ] Show that [Nakayama's lemma](Nakayama's%20lemma.md) may fail if $M$ is not finitely generated.
 - [ ] Show that if $A\in \Alg\slice k$ is finite over $k$ (and an integral domain), then $A$ is a field.
 - [ ] Show that any $M\in\Alg\slice R$ satisfies $M = \colim M_\alpha$ where $\ts{M_\alpha \leq M}$ are all of the finitely generated subalgebras of $M$.
 - [ ] Show that if $M\in \mods{R}$ with $(R, \mfm)$ a local ring, then the action $R\actson M/\mfm M$ factors through the residue field $\kappa(R)$, and $\ts{g_i}\subseteq M$ generate the quotient as an $R\dash$module iff $\ts{a_i + \mfm M}$ generate the quotient as a $\kappa(R)\dash$module.
 - [ ] Show that if $M\in \rmod$ then $M\dual\dual$ is torsionfree, and conclude that not every module is reflexive.

## Integrality

![Exercises in integrality](Unsorted/integrally%20closed.md#Exercises)

## Dimension

 - [ ] Show that a 0-dimensional domain is a field.
 - [ ] Show that a PID is dimension 1 unless it is a field.
 - [ ] Show that a 1-dimensional regular local ring is a DVR.
 - [ ] For $A$ a complete local ring, show that $\dim A = \dim A\complete{\mfm}$
 - [ ] Let $R = \kxn$ and $I = \gens{x_1,\cdots, x_n}$. Show that $\dim R\localize{I} = n$.
 - [ ] Find an infinite-dimensional Noetherian domain.
 - [ ] Show that a regular local ring is integrally closed, but that there are integrally closed local domains of dimension $d\geq 2$ which are not regular.
 - [ ] Show that $R$ is a DVR iff $R$ is a regular local ring of dimension 1.

## Number Theory
 - [ ] Show that ideals satisfy unique factorization in a [Dedekind domain](Dedekind%20domain.md).
 - [ ] Show that a [local](local) [Dedekind domain](Dedekind%20domain.md) is a PID.
 - [ ] Show that PID implies [Dedekind](Dedekind), and all of its [localizations](localizations) are PIDs, [DVRs](DVRs), and thus also [Dedekind domains](Dedekind%20domains).
 - [ ] Show that $\OO_K$ for $K$ an algebraic number field is Dedekind.
 - [ ] Show that [valuation rings](valuation%20rings) are local.
 - [ ] Prove [Hensel's lemma](Hensel's%20lemma.md).
 - [ ] Show that a topological group is Hausdorff if $\ts{0}$ is closed.
 - [ ] Show that for $R \da k[x_1, \cdots ,x_n]$ and $I\da \gens{x_1, \cdots, x_n}$, the $I\dash$adic [completion](completion) is $R\complete{I} = k\formalpowerseries{x_1, \cdots, x_n}$.
 - [ ] Show that
	 - [ ] A PID is a [Dedekind domain](Dedekind%20domain.md).
	 - [ ] For $L/\QQ$ a finite field extension, show that the ring of [algebraic integers](algebraic%20integers) $\OO_L$ is a [Dedekind domain](Dedekind%20domain.md).
	 - [ ] If $A$ is a Dedekind domain with field of fractions $K$ and if $K \subset L$ is a finite separable field extension, then the [integral closure](integral%20closure), $B$, of $A$ in $L$ is a [Dedekind domain](Dedekind%20domain.md).
	 - [ ] A [Unsorted/localization of rings](Unsorted/localization%20of%20rings.md) of a [Dedekind domain](Dedekind%20domain.md) is also a Dedekind domain.
 - [ ] Show that if $R$ is a Noetherian local integral domain whose maximal ideal is principal, then $R$ is a PID and thus a [DVR](DVR.md).
	> Hint: look for a [uniformizer](uniformizer).
 - [ ] Show that in a [Dedekind domain](Dedekind%20domain.md), all [fractional ideals](fractional%20ideals) are [invertible ideals](invertible%20ideals).
 - [ ] Show that if $R$ is a [Dedekind domain](Dedekind%20domain.md) with $\size \spec R < \infty$, then $R$ is a PID.
 - [ ] Show that every ideal in a [Dedekind domain](Dedekind%20domain.md) can be generated by two elements.

## Homological Algebra

 - [ ] Show that $\cocolim^1 M_k = 0$ if $M_k \surjects M_{k+1}$ for all $k$. 
 - [ ] Show that direct limits commute with tensor products.
 - [ ] Show that $\kxn \tensor_k k[x_{n+1}, \cdots, x_m] \cong k[x_1,\cdots, x_m]$
 - [ ] Show that $M\tensor_R \wait$ preserves direct sums.
 - [ ] Prove that $\wait \tensor_R M$ is not generally exact for all $M\in \mods{R}$.
 - [ ] Show that localization is exact.
 - [ ] Show that $M$ is flat iff $\Tor^1(M, R/I)=0$ for all finitely generated ideal $I\normal R$.
 - [ ] Show that $M$ is flat if $\wait\tensor_R M \to M$ is a flat morphism, where it suffices to check on all ideals $I$.
 - [ ] Show that if $R$ is Noetherian and local and $M\in\mods{R}$, then $M$ is free iff $\Tor^1(\kappa(R), M) = 0$ for $\kappa(R)$ the residue field.
 - [ ] Show that over $R=k[x,y]$, the ideals $I\in \Id(R)$ which are projective are precisely the principal ideals.
	 - [ ] Similarly, free ideals in/over a commutative integral domain must be principal.
 - [ ] Show that integral domains have no proper direct summands, and conclude that $R/I$ is never a projective $R\dash$module.
 - [ ] Show that $\gens{x, y} \normal R\da k[x, y]$ is not a free $R\dash$module.
 - [ ] Show the following: a finitely generated module over a noetherian domain is locally free if and only if the dimensions of the fibres over maximal ideals is constant
 - [ ] Show that $\Tor_i^R(k, k) = k$ for all $i$ for $R = k[x]/x^2$.
	Hint: resolve by $1\mapsto x$.

## Flatness

![Exercises](Unsorted/faithfully%20flat#Exercises)

## Numerical Invariants
 - [ ] Prove rationality of the Poincare series of any Poincare series of an additive function on modules.
 - [ ] What is the Poincare series of $A = R[x_0,\cdots, x_n]$ for $R$ an Artin ring?

## Unsorted

 - [ ] Show that for, $M\in \Alg\slice R$ finitely presented implies finitely generated.
	 - [ ] Show that the converse doesn't generally hold, unless $R$ is Noetherian.
	 - [ ] Show that if $M$ is finitely presented, then for all $f$ and $m$, the module of relations $\ker(A^m \surjectsvia{f} M)$ is finitely generated.
	 - [ ] Show that finitely generated projective implies finitely presented.
 - [ ] Show that any ideal Dedekind domain is finitely generated and projective but not free unless they are principal.
 - [ ] Show that for $P$ or $P'$ finitely generated projective, $\Hom(P, P') \cong P\dual \tensor P'$. 
 - [ ] Show that the [going up theorem](going%20up%20theorem.md) fails for $\ZZ \subseteq \ZZ[x]$.
 - [ ] Show that the completions of local rings at non-singular points of a variety over $k$ are all isomorphic.
 - [ ] Give an example of a [local ring](local%20ring) with zero divisors.

## Nakayama

 - [ ] Show the following: if $A$ is a Noetherian local ring with maximal ideal $m \subset A$ and if $m^{n+1}=m^{n}$ then $m^{n}=(0)$. If $A$ is a Noetherian integral domain and $P \subset A$ is a prime ideal then the powers $\left\{P^{n}\right\}, n \geq 1$, are distinct.
 - [ ] Show the following: if $A$ is a Noetherian local ring with maximal ideal $m \subset A$, then
\[
\bigcap_{n \geq 1} m^{n}=(0) \text {. }
\]
If $A$ is a Noetherian integral domain and $P \subset A$ is a prime ideal, then $\bigcap P^{n}=$ (0).
 - [ ] Show that a finitely generated projective module $E$ over a local ring $A$ is free.


![](attachments/Pasted%20image%2020220401022947.png)