# Questions tagged [commutative-algebra]

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

10,809 questions
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### Find an explicit Noether normalization

I am trying to solve the following exercise: Let $k$ be an infinite field and let $f \in k[x_1,...,x_n]\setminus\{0\}$. Define $A = k[x_1,...,x_n, f^{-1}]$ as subring of $k(x_1,...,x_n)$. Find a ...
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### Quotient of free module is still free

Let $f\colon A\longrightarrow B$ be a finite extension of local rings and suppose $f$ flat. We know that under these assumptions (Thm. 2.16 Algebraic Geometry and Arithmetic Curves, Qing Liu ) $B$ is ...
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### Show that this domain is integrally closed

The following is stated on the Wikipedia entry for integrally closed domains as an example: Let $k$ be a field of characteristic not $2$ and $S=k[x_1,...,x_n]$ a polynomial ring over it. If $f$ is ...
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### Question regarding books for commutative algebra

So I was searching about books for commutative algebra. I have read most of the algebra namely galois theory and field theory and basic algebra from dummit Foote. So I was thinking about studying ...
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### Contraction of quotient ideal is quotient of contractions?

Let $\mathfrak a,\mathfrak b$ be ideals in a ring $A.$ The quotient of $\mathfrak a$ and $\mathfrak b$ is $(\mathfrak a:\mathfrak b)=\{x\in A:x\mathfrak b\subseteq \mathfrak a\}$ and if $f:B\to A$ is ...
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### The maximal ideals $\mathrm{Maxspec}(\Bbb Z[X])$ of $\Bbb Z[X]$

In this post, we will try to find all the maximal ideals of $\Bbb Z[X]$, that is $\mathrm{Maxspec}(\Bbb Z[X])$. Of course, there are some posts in MSE or out, but nowhere I found a complete proof. So, ...
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### What is needed to arrive at Bézout's theorem in Algebraic Geometry?

I'm writing a paper that deals with Bézout's theorem, and I'd like to do something from scratch, showing everything I need to get to prove Bézout's theorem and give some examples of this theorem. My ...
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### Dominant projection of affine schemes

Let $A$ be a $k$-algebra of Krull dimension $1$, where $k$ is a field. Let $n\geq 2$ a natural number and $\frak{p}$ a prime ideal of $A^{\otimes n}$ which is not maximal. Consider the ring ...
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### About the concept of a valuation ring

I got a little confused by the different definitions of valuation rings while reading Atiyah's introduction to commutative algebra. Let $A$ be an integral domain and $K$ its field of fractions. We ...
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### Is $A_\mathfrak{p}/\mathfrak{q}_\mathfrak{p}$ an integral domain and is $(A/\mathfrak{q})_\mathfrak{p}$ a ring?

Let $A$ be a commutative ring with identity and $\mathfrak{q}\subset\mathfrak{p}$ two prime ideals of $A$. I am trying to determine whether $A_\mathfrak{p}/\mathfrak{q}_\mathfrak{p}$ is an integral ...
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### Quotient of units in the formal power series ring

Let $k[[x,y]]$ be the ring of formal power series in two variables over a field $k$. A unit in $k[[x,y]]$ is of the form $a_0+f$ where $f\in k[[x,y]]$ and $a_0$ is a unit in $k$. I heard that the ...
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### construction of Cohen Macaulay graph

The graph can be seen in the image Let H be a graph with vertex set V(H)={x_1, x_2,...,x_k, z, w} and J is its edge ideal. Assume that z is adjacent to with deg(z) greater or equal to 2 and deg(w)=1. ...
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### References on Kähler Differentials

I'm working on an independent study project on Kähler differentials for my commutative algebra class. I'm looking for any references on these that might help me out. Any references to their use in ...
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