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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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Is there a consensus on the correct way of raising elements of commutative rings to the power of $a/b$?

I'm trying to understand the "correct" way of raising elements of commutative rings to the power of $a/b,$ where $a$ and $b$ are integers, but not having much luck. Suppose $R$ is a commutative (unital) ring and that $x \in R.$ A first blush attempt…
goblin GONE
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What is the injective hull of a polynomial ring?

The injective hull of a polynomial ring in one variable $K[X]$ (where $K$ is a field) is $K(X)$ since $K(X)$ is a divisible hence injective $K[X]$-module (since $K[X]$ is a PID) and $K(X)$ is an essential extension of $K[X]$. What about more than…
user177523
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Primary decomposition of $(0)$ in $k[X,Y,Z]/(ZY,ZX^2,Z-XY)$

I am looking for a minimal primary decomposition of $(0)$ in $k[X,Y,Z]/(ZY,ZX^2,Z-XY)$. I realize that this is a similar question to some of the previous ones, but the ring is different than in those questions and I would like to both be sure that…
baltazar
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An ideal avoidance

It is known that in a commutative ring $R$ an ideal contained in a finite union of prime ideals $P_i , ( i=1,...,n)$ is a subset of one of them (prime avoidance theorem). Now, if $P_i$'s are arbitrary ideals and $R$ contains an infinite field I want…
karparvar
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Proof of the Jacobian criterion - book of Eisenbud

I could really use some help understanding a statement in the last part of the proof of the Jacobian criterion in "Commutative Algebra with a view toward Algebraic Geometry" by D. Eisenbud, namely: For $R = k[x_1,...,x_n]/I$ we have the usual…
baltazar
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Is $k[x^4,x^3y,xy^3,y^4]$ a local ring?

I noticed that a system of parameters is defined in local rings and some books say that $\{x^4,y^4\}$ is a system of parameters for $R=k[x^4,x^3y,xy^3,y^4]$. Is $R$ a local ring or we use it refers to $R_{\mathfrak m}$? (Here $\mathfrak…
Strongart
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Proving $C\otimes_A\Omega^1_{A/R} \cong \Omega^1_{C/B}$

I am completely stuck on this so any help would be great. Let $R$ be a commutative ring and let $A$ and $B$ be $R$-algebras. Let $C:=A\otimes_RB$. Show that $C\otimes_A\Omega^1_{A/R} \cong \Omega^1_{C/B}$ as $C$-modules.
baltazar
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Localization of a 1 dimensional domain at a maximal ideal is a maximal subring of the fraction field

Given a $1$ dimensional domain $R$ with fraction field $K$. Suppose $\mathfrak{m}$ is a maximal ideal of $R$. Then, I need to show that $R_m$ is a maximal subring of $K$ in the sense that if $x\in K-R_\mathfrak{m}$, then, $R_m[x]=K$. 1)Is this true?…
Timothy Wagner
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Integral closure in field of fractions.

Let $I$ be the ideal generated by $2xy+x^2+y^3$ in $\mathbb{R}[x,y]$. Define $A:=\mathbb{R}[x,y]/I$, I want to find the normalisation of $A$, that is, the set $B= \{ a \in \operatorname{Frac} A : a\text{ integral over }A \}$. What standard methods…
user117449
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If $f:A\to B$ is integral, then $S^{-1}f:S^{-1}A\to S^{-1}B$ is integral?

I'm trying to understand a proof that $f:A\to B$ is integral implies $S^{-1}f:S^{-1}A\to S^{-1}B$ is integral. Here $S$ is a multiplicative subset of $A$. Take $\alpha\in B$, since $B$ is integral over $A$, then there is a…
Waldott
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a very basic question about commutative algebra

Sorry for asking this simple things... First I will put the results that are used. I have a stupid question about the proof of the corollary. I don´t know totally how to use the proposition 1.1, because I can guarantee that there exist a maximal…
August
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showing that the Krull dimension of an extension module is zero

Let $(R,m,k)$ be a Cohen-Macaulay ring of dimension $d>0$ and let $M,C$ be CM $R$-modules such that $\dim M = 0, \dim C = d$. In the proof of Proposition 3.3.3-b(ii) in Bruns & Herzog, the authors write that it is trivial that $\dim…
Manos
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Example of a module $M$ such that $\operatorname{depth}_{\mathfrak p}M<\operatorname{depth}_{A_{\mathfrak p}}M_{\mathfrak p}$; Matsumura, Ex. 16.5

I am looking for an example of a module $M$, a ring $A$, and a prime ideal $\mathfrak p$ such that $\operatorname{depth}_{\mathfrak p} M < \operatorname{depth}_{A_{\mathfrak p}} M_{\mathfrak p}$. How can I find such an example? If $A$ is local…
Lynn16
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Monic polynomials and integral elements.

Let $A \subset B$ be a ring extension, and let $f,g \in B[x]$ be monic polynomials such that $fg \in A[x]$. Prove that the coefficients of $f$ and $g$ are integral over $A$. My attempt was to prove that $A[y]$ is finite (as an $A$-module), for every…
Jack
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a simple question about a local ring, and modules.

I put the paragraph to clarify because it is a vector space. I have a question with the proposition, I don´t know why he concludes the red line assertion, only knowing that there exist a surjective morphism onto $M/mM $ Sorry for my stupid…
August
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