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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.

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Exercise 3.30 from Görtz-Wedhorn: local affine algebras

Here's the Exercise 3.30 from the textbook of Görtz and Wedhorn: Let $k$ be a field, and let $A$ be a local $k$-algebra of finite type. Prove that $\operatorname{Spec} A$ consists of a single point, and that $A$ is finite-dimensional as a…
Sr. Tacuacín
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Characterizing von Neumann regularity

Let $R$ be a commutative unital ring. I want to show the following equivalences: (1) $R$ is zero-dimensional and reduced, (2) every ideal in $R$ is radical, (3) $R$ is von Neumann regular. See wikipedia for the definitions. The only step I am…
ray
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Is the integral closure of a polynomial ring a UFD?

Let $\mathbb{C}(x)$ be the field of rational functions over the complex numbers and $F$ a finite extension of $\mathbb{C}(x)$. Suppose $B$ is the integral closure of $\mathbb{C}[x]$ (the ring of polynomials over $\mathbb{C}$). Is $B$ always a Unique…
vassilis papanicolaou
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What is the relationship between primary decomposition and irreducible decomposition?

Let $A$ be a Noetherian ring and $M$ a finitely generated $A$-module or Noetherian $A$-module. In my commutative algebra class, I was given the following theorems. The first was about the existence of irreducible decomposition of submodules of $M$ :…
Yu Ning
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Modules who have equal support

What can we say about two R-modules, if we know that there supports are equal? What if we know that this modules are Abelian groups? (Z- modules)
David
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Associated prime ideals of $\mathbb C^3$

Let $$A=\begin{pmatrix} 3&2&0 \\0&1&-1\\1&1&1\end{pmatrix}. $$The $\mathbb C$-vector space $\mathbb C^3$ becomes a $\mathbb C[T]$-module via $$\left(\sum_{j=0}^{m}a_jT^j\right)v:=\sum_{j=0}^{m}a_j\left(A^jv\right).$$What are the associated prime…
i.a.m
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Completion (construction Atiyah MacDonald chapter 10)

Following Atiyah MacDonald (Chapter 10: Completions), let $G$ be a topological abelian group. We assume that $0 \in G$ has a fundamental system of neighborhoods consisting of subgroups $G = G_0 \supseteq G_1 \supseteq … \supseteq G_n \supseteq ……
user573597
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Let $R$ be a Noetherian ring and $M$ a finite $R$-module. Show that $\ell(M)<\infty$ if and only if $\text{Supp}(M)\subset m\text{-Spec}(R)$

Let $R$ be a Noetherian ring and $M$ a finite $R$--module. Show that $\ell(M)<\infty$ if and only if $\operatorname{Supp}(M)\subset m\operatorname{-Spec}(R)$. What does $\ell(M)<\infty$ mean? Is it that $M$ is finitely generated? If so, how is it…
Username Unknown
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$pM \neq M$ $\stackrel{?}{\Rightarrow}$ $p M_p \neq M_p$

Let $A$ be a ring, $M$ an $A$-module and $p$ a prime ideal of $A$ such that $pM \neq M$. According to my intuition, it is not necessarily true that $p M_p \neq M_p$. Any counterexample?
Manos
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Flat ring maps and composition

Suppose $A$, $B$ and $C$ are commutative noetherian rings. Suppose we are given maps $f:A\to B$ and $g:B \to C$. Suppose further that $f$ is flat and that $g \circ f$ is also flat. Does it follows that $g$ is flat? Thank you!
the L
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Are these two maps injective? (I'm trying to check a Tor group computation)

Let $R := \mathbb{Z}[x,y,x^{-1},y^{-1}]$. Let $I\subset R$ be the ideal $\langle x+1,y-1,5\rangle$ Let $I'\subset R$ be the ideal $\langle x+1,y-1\rangle$. Let $j : I\hookrightarrow R$ and $j' : I'\hookrightarrow R$ be the natural inclusions. Are…
user355183
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Showing that $I^nM=0$ for some $n$ when $M$ is a finitely generated module and $Supp_A(M) \subseteq \mathcal{V}(I)$

If $M$ is a finitely generated $A$-module, where $A$ is noetherian, and if $I$ is an ideal of $A$ such that $Supp_A(M) \subseteq \mathcal{V}(I)$, I want to show that $I^nM = 0$ for some $n$. An obvious approach is to use the fact that $A$ is…
Wyatt Gregory
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If $P$ is a projective module, then there exist a free module $F$ with $P\bigoplus F$ is free.

I have to prove the above question. So, if $P$ is projective then there exist a $Q$ such that $P\bigoplus Q=F'$ where $F'$ is free. I was thinking that may be we should take $F(Q)-$ free module generated by $Q$ and show that $P\bigoplus F(Q)=F'$.…
uuuuuuuuuu
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Closed, irreducible subset of an affine variety

This question rised when I'm reading Kemper's 'A course in commutative algebra' chapter 6 and 7. let $K$ be a algebraically closed field and $X=\{(x_1,x_2)\in K^2|x_1x_2=0\}$ is an affine variety. The book claimed the height of the maximal ideal in…
scd
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System of Parameters of Local Ring

I have some question (see below) concerning an argument (red question mark) in following example from Bosch's "Algebraic Geometry and Commutative Algebra". Here the excerpt: We consider the ring identification $A = K[t_1,t_2]/(t_2^2-t_1^3) \cong…
user267839
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