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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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The group of invertible fractional ideals of a Noetherian domain of dimension 1

Let $A$ be a Noetherian domain of dimension 1. Let $I(A)$ be the group of invertible fractional ideals of $A$. Let $P$ be a maximal ideal of $A$. Let $I(A_P)$ be the group of invertible fractional ideals of $A_P$. Let $I$ be an invertible fractional…
Makoto Kato
  • 42,602
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a ring of fractions which has finitely many maximal ideals

Let $R$ be a commutative ring and $P_1,\ldots ,P_n$ be prime ideals of $R$. If $S=\bigcap_{i=1}^n (R\setminus P_i)$ then show that the ring of fractions $S^{-1}R$ has only finitely many maximal ideals. The above result will also follow if we can…
pritam
  • 10,157
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Localizations of an integral domain with respect to finite prime ideals

I think the following proposition is likely to be true. I'd like to know a proof of it if any. Proposition Let $A$ be an integral domain, $K$ its field of fractions. Let $P_1, ..., P_n$ be prime ideals of $A$. Let $S = (A - P_1)\cap\cdots\cap(A -…
Makoto Kato
  • 42,602
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Exercise 4.2.17 in Bruns&Herzog (CM Rings)

The exercise states: Let $R$ be a (positive-dimensional, finitely generated) homogeneous algebra over a field $\kappa$. Then there exist integers $a_1\ge a_2 \ge \cdots \ge a_j$, such that \begin{align} P_R(n) = {n+a_1 \choose a_1}+{n+a_2-1…
Manos
  • 25,833
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Wanting to show $a+x$ is a unit for unit $a$ and nilpotent $x$

Possible Duplicate: Units and Nilpotents If $a$ is a unit and $x$ is nilpotent, I'm trying to show that $a+x$ is a unit. Pf.: If $a$ is a unit, there exists a non-zero invertible element $a^{-1}$ s.t. $a\cdot a^{-1} = 1$, and if $x$ is nilpotent…
James R.
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Why do the mapping from $A^m$ to $M_n$ is surjective?

I was unable to solve the problem 1.3.11(b) from Qing Liu's Algebraic Geometry and Arithmetic Curves. Let $A$ be a commutative complete ring for the $I$-adic topology with unit, where $I$ is an ideal of $A$. Let $(M_n)_{n\geq 0}$ be $A$-modules such…
studying
  • 31
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Showing the Dimension of a Ring

Let $f$ be in $\mathbb{Z}[x,y]$ and consider the the quotient ring $\mathbb{Z}[x,y] / \langle f \rangle$. The ring $\mathbb{Z}[x,y]$ has dimension 3, and the codimension of $\langle f \rangle$ is $\le 1$ by Krull's Ideal Theorem. Is it possible to…
Paul Slevin
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Why is $(bc,a_2,\ldots,a_n)$ an $M$-sequence if $(b,a_2,\ldots,a_n)$ and $(c,a_2,\ldots,a_n)$ are?

Let $R$ be a commutative Noetherian ring and $M$ be a nontrivial finitely generated $R$-module. Suppose $(b,a_2,\ldots,a_n)$ and $(c,a_2,\ldots,a_n)$ are $M$-sequences. The following fact is given: if $m_1,\ldots,m_n,m_1',\ldots,m_n'\in M$ are such…
happyEddie
  • 362
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Can we find an ideal $I$ s.t. $ k[x,y]/I$ = $k[x^2,y^2,xy]?$

Can we find an ideal $I$ of $k[x,y]$ s.t. $ k[x,y]/I$ = $k[x^2,y^2,xy]?$
user198206
3
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1 answer

Vanishing of Ext group and Krull dimension

Suppose $A=k[x_1,..,x_n]_{(x_1,..,x_n)}$, it is a regular local ring of dimension $n$. Let $B=A/I$ be a quotient ring of Krull dimension $r$. How to show $\operatorname{Ext}_A^i(B,A)=0$ for $i
user93417
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1 answer

Not a radical ideal

Is there a method to compute the radical of an ideal? for example take $J=(xw-y^{2},xw^{2}-z^{3}) \subset k[x,y,z,w]$. I want to show $J$ is not radical, I guess the idea is to add and substract terms until we get something which is in the radical…
user10
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Proof of a lemma which leads to Nakayama's lemma

I am trying to understand the proof of the following statement: Let $A$ be a commutative ring, let $M$ be a finitely generated $A$-module and $I$ an ideal of $A$ such that $IM=M$. Then there is an $a\in I$ such that $(1-a)M=0$. Proof. If $M=0$,…
Jimmy R
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The proof of Krull's Principal Ideal Theorem

Theorem: Let $R$ be Noetherian and $P$ be a minimal prime ideal over $(a)$ for some nonunit $a$ of $R$. Then $\operatorname{ht}(P)\leq 1$. My lecture notes prove this as follows. WLOG $R$ is local with unique maximal ideal $P$. Show $R/(a)$ is…
user3131035
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Is an irreducible element still irreducible under localization?

Suppose $R$ is a domain. We say an element $x\in R$ is "irreducible" if $x=yz$ implies that $y$ or $z$ is a unit or both are units. I want to know if an irreducible element is still an irreducible element when taking localization. If not, what is…
wxu
  • 6,671
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A proof for Atiyah-Macdonald Exercise I.21.iii

The following is exercise I.21.(iii) of Atiyah-Macdonald: Let $\phi \colon A \to B$ be a ring homomorphisms. Let $X = \operatorname{Spec} A$ and $Y = \operatorname{Spec} B$ [and let $\phi^\ast \colon Y \to X$ be the induced mapping, $\phi^\ast…
Earthliŋ
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