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Questions tagged [commutative-algebra]

Questions about commutative rings, their ideals, and their modules.

Commutative algebra is the area of mathematics that deals with commutative rings and their ideals, as well as modules over commutative rings.

Many results and tools of commutative algebra are cornerstones of algebraic geometry. Important tools of commutative algebra include localization and completion of rings and modules.

16857 questions
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Hom and tensor with a flat module

Let $A$ be a commutative noetherian ring. Let $M, N$ be $A$-modules, and assume that $M$ is finite over $A$. Let $P$ be a flat $A$-module. Is it true that there is an isomorphism $\operatorname{Hom}_A(M,N)\otimes_A P \cong…
the L
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Birational and faithfully flat $\implies$ isomorphism

Let $A \subseteq B$ be integral domains with the same field of fractions. Assume that $A \to B$ is faithfully flat. Why do we have $A=B$? This is an exercise in Matsumura's book. Here is my idea: If $b \in B$, consider $I = \{a \in A : ab \in A\}$.…
Martin Brandenburg
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Characterization of faithfully flat homomorphisms

Let $A \to B$ be a homomorphism of commutative rings. Why are the following conditions equivalent? $A \to B$ is faithfully flat. $A \to B$ is injective, flat and $B/A$ is a flat $A$-module. This should be elementary, but at the moment I don't see…
Martin Brandenburg
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Finitely generated ideals in a Boolean ring are principal, why?

The classical book on commutative algebra Introduction to Commutative Algebra, by Atiyah and Macdonald, has the following as exercise I.11. A ring is Boolean if $x^2=x$ for any $x$ of $A$. In a Boolean ring $A$, show that i) $2x=0$ for all…
awllower
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A proof that shows surjective homomorphic image of prime ideal is prime

Let $A, B$ be commutative rings with $1_{A}, 1_{B}$. Suppose that $\mathfrak{p} \neq (1)$ is a prime ideal in $A$ with $\mathfrak{p} \supseteq \ker{\varphi}$ where $\varphi: A \rightarrow B$ is a surjective homomorphism. I want to show…
user123454321
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Finitely generated algebra

I am getting the confusion with the definition of algebra. When we say $A$ is a finitely generated $R$- algebra then is that mean $A$ has a ring structure and finitely generated as an $R$-module. Thanks
Rajesh
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When is the integral closure of a local ring also a local ring?

Suppose $A$ is a normal local domain contained in a field $K$. Suppose $B$ is the integral closure of $A$ in $K$. Under what conditions on $A$ is $B$ local?
Moosa Pei
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How badly can Krull's Hauptidealsatz fail for non-Noetherian rings?

Krull's Hauptidealsatz (principal ideal theorem) says that for a Noetherian ring $R$ and any $r\in R$ which is not a unit or zero-divisor, all primes minimal over $(r)$ are of height 1. How badly can this fail if $R$ is a non-Noetherian ring? For…
Zev Chonoles
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Extending a homomorphism of a ring to an algebraically closed field

Let $A$ be a subring of a ring $B$. Suppose $B$ is integral over $A$. Let $\Omega$ be an algebraically closed field. Then every homomorphism $\psi\colon A \rightarrow \Omega$ can be extended to a homomorphism $\phi\colon B \rightarrow \Omega$?
Makoto Kato
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$K[[X]]$ is not a finitely generated $K[X]$-module.

How can I prove that $K[[X]]$ is not finitely generated over $K[X]$ as a module, where $K$ is a field. What I tried: if above is not true then $K[[X]]$ is integral extension over $K[X]$. But I failed to draw any contradiction. Help me. Thanks
user185640
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An ideal is homogeneous if and only if it can be generated by homogeneous elements

Let $S$ be a graded ring with decomposition $S = \bigoplus_{d \geq 0} S_d$, where the $S_d$ are additive abelian groups such that $S_d S_e \subseteq S_{d+e}$ for $e,d \geq 0$. An element in $S_d$ is called a homogeneous element of degree $d$. An…
Matt
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Are projective rings over $\mathbb{Z}$ free?

Let $R$ be a ring. Say that an $R$-algebra $A$ is $R$-projective if it has the left lifting property with respect to surjections of $R$-algebras: that is, whenever $B \to C$ is a surjection of $R$-algebras, then $\hom_R(A, B) \to \hom_R(A, C)$ is a…
Akhil Mathew
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Fraction field of the formal power series ring in finitely many variables

What is the fraction field of the formal power series ring over a field in finitely many variables $K[[X_1,\dots,X_n]]$? Is there a nice description for this field? When $n=1$, I know this is the formal Laurent series ring over $K$.
Marcus Lode
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Exercise 3.15 [Atiyah/Macdonald]

I have a question regarding a claim in Atiyah, Macdonald. A is a commutative ring with $1$, $F$ is the free $A$-module $A^n$. Assume that $A$ is local with residue field $k = A/\mathfrak m$, and assume we are given a surjective map $\phi: F\to F$…
Sam
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Homomorphism of local rings

Let $(A, \mathfrak{m})$ and $(B, \mathfrak{n})$ be local Noetherian rings. Suppose that $\phi \colon A \rightarrow B$ is a map such that $\phi(\mathfrak{m}) \subset \mathfrak{n}$ and suppose $A/\mathfrak{m} \cong B/\mathfrak{n}$; $\mathfrak{m}…
Honghao
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