All Questions
Tagged with ideals commutative-algebra
1,564
questions
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Equality of two completions
I have the following question.
Suppose $R$ is Noetherian ring, $I$ is ideal in $R$ and $S$ is multiplicatively closed set. Let $(I^n\colon\langle S\rangle) = \varphi^{-1}(I^nS^{-1}R),$ where $\varphi\...
- 85
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Are determinantal ideals Cohen-Macaulay?
Let $R=K[X_{ij}:i=1,\dots,m,j=1,\dots,n]$. The ideal in $R$ generated by all the $t$-minors of the $m\times n$ matrix
$$ X=\begin{pmatrix}
X_{11} & X_{12} & \dots & X_{1n}\\
X_{21} &...
- 185
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Set of zero-divisors in a ring is a union of prime ideals [duplicate]
This is a problem from Atiyah-MacDonald Introduction to Commutative Algebra and it goes as follows:
In a ring $A$, let $\Sigma$ be the set of all ideals in which every element is a zero-divisor. Show ...
- 557
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Understanding Radicals as Intersections of Prime Ideals and as the Preimage of a Nilradical
I ran into a proposition that the radical $r(a)$ equals the intersection of the prime ideals containing $a$. Then it is said that the latter can be understood through the following result: Let $R$ be ...
- 557
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If $R$ is an integral domain, $I$ is an ideal of $R$, and $0\neq f: I \to R$ is an $R$-module homomorphism, can we conclude that $f$ is injective?
If $R = \mathbb{Z}$, $0 \neq I \unlhd \mathbb{Z}$, and $0 \neq f: I \to \mathbb{Z}$ is an arbitrary $\mathbb{Z}$-module homomorphism, then $f$ must be injective.
This leads to the question:
If $R$ is ...
- 923
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Is $A/I\otimes_C B/J$ always isomorphic to $A\otimes_C B/\langle I\otimes 1,1\otimes J\rangle$?
Let $A$ and $B$ and commutative $C$-algebras, then in what cases is the tensor product $A/I\otimes_C B/J$ isomorphic to the ring $A\otimes_CB/\langle I\otimes 1,1\otimes J\rangle$? It seems this holds ...
- 3,431
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Proof of a method of calculating the ideal quotient $I:\langle f^{\infty}\rangle$
Background:
(1)Let $I$ be an ideal of $K[x_1...x_n]$,and let $f\in K[x_1...x_n]$,where K is a field.
We define $I:f^{\infty}:=\bigcup_{i=1}^{\infty}I:\langle f^i\rangle$.
And I have known that $\...
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57
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Is $\langle x+y^3,x^3+y \rangle$ a radical ideal in $\mathbb{C}[x,y]$?
Let $u=x+y^3, v=x^3+y$.
Question: Is it true that $I=\langle u,v \rangle$ is a radical ideal?
Of course, $I$ is not a maximal ideal, since its generators have more than one common zero; the common ...
- 6,705
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Pathologies with Product Ideal $I \cdot J $
Let $R$ be an unitary, commutative, Noetherian ring and $I, J$ two ideals.
The product ideal $I \cdot J$ is defined as
$$ I \cdot J:=\{\sum_i^n a_i \cdot b_i \mid a_i\in I,b_i\in J\}$$
note we allow ...
- 7,589
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41
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Prove that $V(I(X)) = X$
Let $A$ be a ring and $X \subset SpecA$ a closed set.
Prove that $I(X) := \bigcap P \in XP$ is a radical ideal.
My proof is as follows:
$I(X)$ is an ideal since it's the intersection of ideals. If $f^...
- 101
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45
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I am getting very confused by the definition of a minimal prime ideal
I am trying to prove that in an affine scheme $\operatorname{Spec}A$ that an irreducible component can be written as the vanishing locus $V(\mathfrak p)$ for a minimal prime ideal $\mathfrak p$, but I ...
- 3,431
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36
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Maximal m-system vs zero-divisors
Let $R$ be a commutative nilpotent-free ring with unity and $\mathfrak{M}$ be a maximal $m$-system. I want to show the following.
For any nonzero $a\notin \mathfrak{M}$; there exists $b\in \mathfrak{...
- 1,078
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If $f: A \to B$ is a homomorphism, then a prime $\mathfrak{p} \subseteq A$ is the contraction of a prime in $B$ iff $\mathfrak{p}^{ec} = \mathfrak{p}$
This is proposition 3.16 in Atiyah-Macdonald, and I'm a little confused at the proof. Their notation is $S = f(A \setminus \mathfrak{p})$, and $\mathfrak{p}^{ec}$ means to extend the prime ideal then ...
- 137
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Show that the kernel of the homomorphism $\varphi: k[X, Y, Y/X]\to k[x, y, y/x]$ is prime of height $1$.
The problem arises from reading this answer.
The Problem: Let $k$ be a field, let $R=k[X, Y]/(Y^2-X^3+X^2)$. Henceforth use lowercase letters to denote the elements in the quotient ring (e.g., $x=\...
- 1,467
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Long exact sequence of intersected prime ideals
I'm considering a commutative local noetherian ring $R$ (in fact, $R$ is even Cohen-Macaulay) along with a number of prime ideals $I_i$. I can then construct the exact sequence
\begin{equation}
0\...
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