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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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A one-dimensional domain with infinitely many maximal ideals and non-zero Jacobson radical?

I'm looking for an example of a non-semi-local, non-Jacobson domain $A$, having $\dim(A)=1$. A commutative ring $A$ is non-Jacobson if it has a prime ideal that is not an intersection of maximal ideals. For a one-dimensional domain, that comes down…
Matthé van der Lee
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A question about integral domains and inclusion relations in abstract algebra

Let $R$ be a Prüfer domain with quotient field $K$ and $\sum$ be the set of all semilocal Prüfer domains $R'$ with $n$ maximal ideals and quotient field $K$ such that $R\subseteq R'$. Let $R=R'_{1}\cap\ldots\cap R'_{t}$ where $R'_{1},\ldots,R'_{t}$…
Mary
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Maximal Ideals in $K[X,Y]$

So, given a field $K$ and a polynomial $f(x,y)$ in the ring of polynomials $K[X,Y]$, I am trying to understand why the following statement is true: If $f(a,b) = 0$ for $(a,b) \in K \times K$, then $(f) \subset (x-a,y-b)$. I know that roots for a…
Commutative Algebra
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Determining if a quotient of $\Bbb Q[t_1, t_2, \ldots]$ is Noetherian

Let $L = \Bbb Q[t_1, t_2, \ldots]$ (polynomial ring in infinitely many variables). Let $I$ be the ideal of $L$ generated by $t_1^2$ and $t_i - t_{i+1}^2$ for all $i$. I am allowed to assume that $t_1 \notin I$. The question is, is $R = L/I$…
pizzaroll
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Modules over local ring and completion

I'm stuck again at a commutative algebra question. Would love some help with this completion business... We have a local ring $R$ and $M$ is a $R$-module with unique assassin/associated prime the maximal ideal $m$ of $R$. i) prove that $M$ is also…
Dquik
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A subring of $k[x,y]$ is not Noetherian even if $k$ is finite

It was an exercise in Basic Algebra by Cohn: If $k$ is infinite field then the subring of $k[x,y]$ given by $k+xk[x,y]$ is not Noetherian. Proof: Consider following chain of ideals in $k+xk[x,y]$: $$(xy) \subset (xy,xy^2) \subset…
Beginner
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Localization of an injective module is also injective

The question is very simple: Do you know an easy proof of the following: Let $I$ be an injective $R$-module and $a\subset R$ an ideal. Then the localized module $ I_a$ is again injective. Of course we are working in commutative algebra. I know a…
Miguel
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Proof of Atiyah Macdonald's Introduction to Commutative Algebra - Corollary 2.7 and Proposition 2.8

These are the propositions in question. I can't understand the observations for $\alpha(M/N) = (\alpha M+N)/N$ and $N+mM = M$. In general how do we prove observations like this. I can't think beyond looking for a homomorphism and then quotienting…
Arjun Sinha
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Study of a map between completions

Let be R a ring and let $f:R[\! [ X_1, ..., X_n ]\! ]\rightarrow R[\! [ X_1, ..., X_n ]\! ]$ an homomorfism of $R$-algebras. Called $J\in M_{n\times n}(R)$ the jacobian matrix $J=\left( \frac{\partial f_i}{\partial X_j}(0)\right)$ where…
Acuo95
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Minimal number of generators of an ideal in a Noetherian local ring

I have seen the phrase "minimal number of generators of an ideal" (in a Noetherian local ring) several times. I am unable to see how this is a well defined. Explicitly, how do we show, if $x_1,...,x_m$ and $y_1,...,y_n$ are minimal generating sets…
Yan Etor
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Akizuki counterexample in Reid's Undergraduate Commutative Algebra

In his book Undergraduate Commutative Algebra Miles Reid describes example of Noetherian ring such that its normalisation (integral closure in field of fractions) is not finite ring extension. He takes $k$ to be a field, and $K$ an algebraic field…
user468985
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Every radical ideal in $k[x_1,...,x_n]$ is contained in at least $2$ maximal ideals

Assume that $I$ is a proper radical ideal of $k[x_1,...,x_n]$ which is not a maximal ideal where $k$ is a field. Prove that $I$ is contained in at least two maximal ideals. Now it is easy to prove the claim using Hilbert's Nullstellensatz. I wonder…
Levent
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Divisorial ideal of a Krull domain

Now I try to do exercise 12.4 in the book "Commutative ring theory" by H. Matsumura. Let $A$ be a Krull domain, $I\subseteq \mathfrak p$ and $\mathfrak p$ is a height $1$ prime ideal of $A$. I don't know how to prove the following…
yaxuan
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Primary decomposition of ideal saturation

This is a follow-up to my question here: Primary decomposition and contraction Let $R$ be a Noetherian ring, $I\subset R$ ideals, and $S \subset R$ a multiplicative set. Define: $$I:\langle S\rangle := (IR_S)\cap R = \{ x \in R \mid xy \in I \text{…
walkar
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Proving this algebra fact

The Nullstellensatz for $\mathbb{C}[x_1, \ldots, x_n]$ gives a dictionary between radical ideals and varieties, which makes the following assertion obvious: a radical ideal in $\mathbb{C}[x_1, \ldots, x_n]$ is the intersection of the maximal ideals…
Tony
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