Questions tagged [primary-decomposition]

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Is $S^{-1}(q_i)$ is a primary ideal in $S^{-1} A$?

This question was left as an exercise in class of commutative algebra and I am struck on it. Let A be a noetherian ring and $I=\cap_{i=1}^n q_i$ be a minimal primary decomposition. Let $p_i = \sqrt{...
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A question in 1st uniqueness theorem of primary decomposition

I am self studying commutative algebra from a class notes based on atiyah and macdonald and I am struck on this proof. Statement; Let $I=\cap_{i=1}^n q_i$ be a minimal primary decomposition of I. ...
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Finding the kernel of a given map

Let $\mathbb K$ be an algebraic closed field and let $f,g\in\mathbb K[X,Y]$ be two polynomials so that $V_{\mathbb K}(f)$ and $V_{\mathbb K}(g)$ don't have common irreductible components and $(0,0)\in ...
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Associated prime of certain tensor product

Let $A$ be a Noetherian ring and $p\subset A$ a prime ideal. Is $p\otimes_A A/p$ a torsion-free $A/p$-module? The case which I'm interested in is $A:=k[X,Y,Z]/(X^2-YZ)$ and $p:=(x,y)$, where $k$ is a ...
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Examples of primary decomposition of a submodule

Theoretically primary decomposition can be defined for submodules, but all examples that I've seen were for ideals (except modules over PIDs, but they are completely elementary). Are there any ...
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Concerning a minimal polynomial

I'm currently taking linear algebra and I'm working on a questing concerning minimal polynomial. The question is as follows: Consider a matrix $A$, whose minimal polynomial is of the form $p(x)^n$ ...
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A different proof of exercise 4.6 from Atiyah-Macdonald

I know that this problem has already been solved a couple of times but I was here because while going through the exercises I had written a slightly different proof and since I have no one to check it,...
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Primary decomposition of a radical ideal [duplicate]

Let $R$ be an arbitrary (commutative) ring. Let $I \subseteq R$ be an ideal such that $I$ has a minimal primary decomposition $$I =\bigcap_{i=1}^s Q_i$$ with $Q_i$ $P_i$-primary ideal, $i=1, \dots,s$....
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Exercise about saturations of a ring

Let $A$ be a ring, $S$ any multiplicatively closed subset. For an ideal $I$, let $S(I)$ denote the contraction of $S^{-1}I$ in $A$. If $I$ has a primary decomposition, show that the ideals of the form ...
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Associated primes of $(0)$

Let $A$ be a ring, and let $\bigcap_{i=0}^rq_i=(0)$ be a minimal primary decomposition of the zero ideal. Define $D\subseteq \operatorname{Spec}A$ as the set of the prime ideals for which exists an $a\...
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Exercise 4.2, Atiyah Macdonald [duplicate]

I'm having difficulties in showing that a radical ideal has no embedded primes. I know that a radical ideal $I\subseteq A$ ($A$ ring) is equal to the intersection of the minimal primes containing $I$. ...
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Stuck with following question on primary decomposition theorem.

Let $T$ an operator over a $\mathbb{F}$ vector space $\mathbb{V}$, with $\dim(\mathbb{V})<\infty$. Let $p=p_{1}^{r_{1}}\cdots p_{k}^{r_{k}}$ the minimal polynomial of $T$, and $\mathbb{V}=W_{1}\...
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Showing that a primary ideal is minimal

Like in my previous question (Exercise about primary ideals), consider the canonical homomorphism $A\to A_\mathfrak p$, for a fixed prime ideal $\mathfrak p$ in the ring $A$, and write $I\cap A$ for ...
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Running a minimal primes decomposition in Macaulay2

I want to compute the minimal primes decomposition of a very complicated ideal using Macaulay2. My computer's 16gb of RAM isn't enough to run the code without crashing, but I already know 46 of the ...
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Some characterizations about T-indecomposable vector spaces

Recently I have found this statement and I would like to prove as exercise. Let $V$ be a vector space with $dim_{\mathbb{K}}(V)=n \in \mathbb{N}$ and $T\in End(V)$. The following facts are equivalent:...
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If $x\in I$ is such that $\text{Supp}(0:_M x)\subseteq V(I)$ , then how to show that $x \notin \cup_{P\in \text{Ass}_R(M)\setminus V(I)} P $?

Let $M$ be a finitely generated module over a Noetherian ring $R$. Let $I$ be an ideal of $R$. Since Ass$_R (M)$ is finite, and $I\nsubseteq P$ for all $P\in$Ass$_R (M)\setminus V(I)$, so by prime ...
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Hoffman Kunze Section 6.8 Exercise 12

In the Linear Algebra book by Hoffman Kunze, Exercise 12 of section 6.8 goes If you thought about Exercise 11, think about it again, after you observe what Theorem 7 tells you about the ...
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A zero-divisor whose every localization is zero in a Noetherian ring whose every associated prime is minimal

Let $R$ be a commutative Noetherian ring such that all associated primes are minimal primes. Let $r\in R$ be a zero-divisor such that $r/1=0$ in $R_P$ for every minimal prime $P$ of $R$. Then, is it ...
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Compute $\text{Ass}(R/p^2)$

Problem: Consider the ring $R=\mathbb{Q}[X,Y,Z]/\langle{XY-Z^2}\rangle$, and the ideal $p=\langle{x,z}\rangle$, where $x,y,z$ are the residues of $X,Y,Z$. Show that ($1$) $p$ is prime; ($2$) $p^2$ is ...
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For which Noetherian local rings, is every associated prime ideal contained in the square of the maximal ideal?

Let $(R,\mathfrak m, k)$ be a Noetherian local ring of depth $>1$. If $a\in R$ is a zero-divisor, i.e. if $ab=0$ for some $0\ne b \in R$, then must it be true that $a\in \mathfrak m^2$? Since the ...
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Homological proof of $𝑝$-primary decomposition of torsion abelian groups

It is known that torsion abelian groups decompose as direct sums of their $𝑝$-primary components. Is there any homological proof of this fact using derived functors? It can be proved by hand but I ...
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Proof verification for primary decomposition

Let $R=\Bbb{C}[x,y,z]/(xy-z^2)$ and $I=(x,z)$. Find the primary decomposition of $I^2$. So one can verify that since $I=(x,z)$ then $I^2=(x^2,z^2,xz)$, and since $z^2=xy$ we can conclude that $I^2=(x^...
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About the height of primes associated to a squarefree monomial ideal $I\subseteq J$

Edit Let $S=K[x_1,\dots,x_n]$ be a polynomial ring in $n$ indeterminates with coefficients in a field $K$. For a monomial ideal $I$ of $S$, $G(I)$ denotes the minimal generating set of $I$. For ...
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Hoffman and Kunze Linear Algebra Section 6.8 Decomposition Theorem

Below you can find the related theorem and part of the proof that I am confused about. I understand that with the related definition of $E_i$'s we get k linear operators on $V$that satisfy $E_1+...+...
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Counterexamples to standard assertions on associated primes

I have recently started reading section 6 of Matsumura's "Commutative Ring Theory", and I have noticed that some of the results from Theorems 6.1 through 6.5 make the assumption of ...
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Understanding why this linear operator is diagonalizable.

let me show you the context of my question: Let $T:V \to V$ a linear operator, where $V$ is a finite dimensional vector space and $m_t(x)=(x - \lambda_1)^{m_1} \cdots(x-\lambda_k)^{m_k}$. In the proof ...
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every ideal in a Noetherian ring is an intersection of ideals primary to maximal ideals?

I'm reading a lecture note Tight Closure of Huneke and he stated that "Every ideal in a Noetherian ring is an intersection of ideals primary to maximal ideals". I don't see why this is the ...
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Give an example of a proper ideal I of a ring R such that the radical of I is equal to R i.e. √I=R.

I am having trouble with finding such example. In commutative case with identity such example does not exist as 1∈R=√I implies 1∈I and so I=R. Is there any in noncommutative case or a ring without ...
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1 vote
1 answer
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Find minimal primary decompositions of ideals

(1) In $R = \mathbb{R}[x,y]/(x^2 + y^2 -1)$, find a minimal primary decomposition of $(\bar{x}^2)$. (2) In $R = \mathbb{Z}[\sqrt{-5}]$, find a minimal primary decomposition of $(6)$. For (1), I know ...
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Finding equidimensional radical of an ideal

In the article by $\textit{Eisenbud D., Huneke C., Vasconcelos W.}$ titled as $\textit{Direct Methods for Primary Decomposition (1992, Inventiones Mathematicae,110, pg.207 - 235)}$ Proposition 2.3 is ...
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Example of Ascending Chain Condition(ACC) on annihilator

$(x) \supseteq (x^2) \supseteq (x^4) ... $ is a descending chain in $A[x]$ that is not stationary if $A[x]$ is not Artinian. What is an example of a module such that $Ann(x) \subseteq Ann(x^2) \...
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About primary decomposition in the Graded ring

Let $R$ be a graded Noetherian ring and $M$ be f.g. graded $R$-module. What I want to show is that Primary decomposition of submodule of $M$ can be taken in terms of homogeneous modules. (In fact it ...
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Elements of a Module Whose Annihilator is a Given Associated Prime

Given a commutative ring $R$ and an $R$-module $M$, an associated prime of $M$ is a prime ideal $\mathfrak{p} \subset R$ such that $\mathfrak{p} = \operatorname{Ann}(x)$ for some $x \in M$. I'm ...
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Proposition 4.7 in Atiyah-Macdonald

In Atiyah/Macdonald's Introduction to Commutative Algebra, the following proposition is given (4.7): Let $\mathfrak{a}$ be a decomposable ideal in a ring $A$, let $\mathfrak{a} = \bigcap_{i=1}^n\...
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One dimensional noetherian domain $\implies $ uniqueness of primary decomposition of ideals?

Let $A$ be a Noetherian integral domain whose Krull dimension equals 1. Given $I\subseteq A$ any ideal, prove or give a counter example: the minimal primary decomposition of $I$ is unique. If $I = \...
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Primary Decomposition Theorem; proof?

I am a self learner and have attempted to prove the Primary Decomposition Theorem below. Any help either saying it's correct or highlighting something that's wrong or any misunderstanding would be ...
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2 questions in Statement of Primary Decomposition Theorem and it's Corollary in Linear Algebra

I am self studying Linear Algebra from Hoffman Kunze and I have 2 questions in section 6.8 whose image I am adding-> Questions : (1) In the last paragraph how does one can deduce that $W_{i}'s $ ...
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Primary decomposition of an ideal and its extension

I'm trying to solve a problem in Sharp's Steps in Commutative Algebra, to be precise Exercise 4.22 which states the following: Let $f:R \rightarrow S $ be a surjective homomorphism of commutative ...
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A question about a step in a proof of the Krull Intersection Theorem

Lately, I have been using Steve Kleiman and Allen Altman lecture notes on commutative algebra, A Term of Commutative Algebra, that are available for free on internet, to study the subject. In those, ...
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show that a matrix is similar to a block matrix using minimal polynomial

$$\begin{pmatrix}1&0&-4&4\\ \:\:\:\:0&2&0&0\\ \:\:\:\:0&1&1&0\\ \:\:\:\:0&1&0&1\end{pmatrix}$$ I need to find a matrix $P$ where $PAP^{-1}=D$ where $D=\...
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Localisation map is zero or injective, then zero ideal is primary.

Question Let $M$ be an $R$ module such that for every multiplicative closed set $S\subset R$, the kernel of natural map $M\to S^{-1}M$ is $(0)$ or $M$. Show that $(0)$ is primary in $M$. Attempt I ...
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Find the associated primes of $\mathbb{C}[x]/{\langle x^{3}+1 \rangle}$

This intuitively to me seems to be polynomials of degree 2 or less. My idea was to show that $\langle x^{3}+1\rangle$ is primary and as $\mathbb{C}[x]$ is Noetherian we have the assassin as $\text{Ass}...
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2 votes
1 answer
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Proof that all ideals in a Noetherian ring have a primary decomposition?

For each proof relating to this there always seems to be a part where they assume without loss of generality that the ideal they are trying to find the primary decomposition of is the zero ideal. For ...
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2 votes
1 answer
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Solve using the Laplace transform of the initial value problem

The given equation and initial values are: $$\frac{d^2x}{dt^2}+25x=50e^{5t}$$ $$x(0)=0 \space, x^{'}(0)=0$$ Then taking the Laplace transform of the given: $$\mathscr{L}\left[x^{''}+25x \right]=\...
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Decomposing an inverse Laplace transform

The given inverse Laplace transform is: $$\mathscr{L}^{-1}\left[\frac{5s^2+12s-4}{s^3-2s^2+4s-8} \right]$$ First split it up into three separate fractions and factorize the denominator $$\mathscr{L}^{...
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  • 1,063
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Non-uniqueness of embedded primary components

I'm studying the Uniqueness theorems for primary decomposition of decomposable ideals in a ring. In my notes, after having proved the two classical theorems, there is the following remark: Remark. ...
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On the intersection of integral closure of powers of an ideal

Let $R$ be a commutative Noetherian ring. Let $J$ be a proper ideal of $R$ . Let $Min(R)$ be the set of minimal primes of $R$ (this set is finite as $R$ is Noetherian). Then how to prove that $\cap_{...
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On the first Local Cohomology module of a complete local ring of depth $1$

Let $(R,\mathfrak m)$ be $\mathfrak m$-adically complete Noetherian local ring of depth $1$. Thus the local cohomology module $H^1_{\mathfrak m}(R)$ is a non-zero Artinian module. My question is: How ...
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Symbolic Rees Algebra of an ideal in a Noetherian excellent ring

For an ideal $I$ in a commutative Noetherian ring $R$ and integer $n\ge 0$, the $n$-th symbolic power of $I$ is define as $I^{(n)}:=\cap_{P\in Ass(R/I)} \phi_P^{-1} (I^nR_P)$ , where $\phi_P : R\to ...
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Zero Ideal is Indecomposable as a Finite Intersection of Primary Ideals on an Infinite Ring of Idempotents

Let R be a ring of idempotents, that is, a ring in which holds $x^2=x, \forall x \in R$. Prove that if R is infinite, then (0) is indecomposable as a finite intersection of primary ideals. Any hints ...
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