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Questions tagged [primary-decomposition]

For questions related to primary decomposition. In mathematics, every ideal can be decomposed as an intersection, called primary decomposition, of finitely many primary ideals (which are related to, but not quite the same as, powers of prime ideals).

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Completion of primary ideal is primary

I have the following question. Suppose $(R, \mathfrak{m})$ is a local Noetherian ring, $\mathfrak{a}$ is $\mathfrak{p}$-primary ideal. Is it true that $\hat{\mathfrak{a}}$ is a primary ideal in $\hat{...
abcd1234's user avatar
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Problem of finding the invariant factors and elementary divisors of a quotient group.

$\mathbf{The \ Problem \ is}:$ Let $G:= \Bbb{Z}_9\times \Bbb{Z}_9\times \Bbb{Z}_9$ and $H:= \langle(6,6,6)\rangle.$ Find the invariant factors and elementary divisors of the quotient group $G/H.$ $\...
Rabi Kumar Chakraborty's user avatar
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$\text{depth }A\leq \text{depth }_{\frak{p}}A + \dim(A/{\frak{p}})$, where $A$ is a Noetherian local ring.

This is Exercise 16.5 in Commutative Algebra, Matsumura or Exercise 17.5 in Commutative Ring Theory by the same author. Let $(A,\frak{m})$ be a Noetherian local ring and $\frak{p}$ be a prime ideal. ...
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Geometric explaination of $I=\sqrt{I}$ implies $I$ has no embedded prime ideals

I'm working on the following exercise of Atiyah Commutative Algebra: Let $I$ be an ideal of a ring $A$ with $I=\sqrt{I}$ and has a primary decomposition. Show that $I$ has no embedded prime ideals. ...
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Under Noetherian assumptions, the associated primes equal the primes belonging to the zero submodule (Exercise 7.18 of Atiyah, Macdonald)

In Atiyah, Macdonald, Introduction to Commutative Algebra, Chapter 7, we find: 7.18. Let $A$ be a Noetherian ring, $\mathfrak{p}$ a prime ideal of $A$, and $M$ a finitely generated $A$-module. Show ...
Elías Guisado Villalgordo's user avatar
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1st and 2nd uniqueness theorems for primary decomposition of modules

In Chapter 4 of Atiyah, Macdonald, Introduction to Commutative Algebra, we find the following exercises: 4.21. An element $x \in A$ defines an endomorphism $\phi_x$ of $M$, namely $m \mapsto x m$. ...
Elías Guisado Villalgordo's user avatar
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Finding a primary decomposition of an ideal.

I'm studying commutative algebra and I need to find, for the course I'm attending, the primary decompositions of the following ideals, with some extra questions: In $\mathbb{C}[x,y]$, find, if you ...
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Modules and Diagonizability using Primary Decomposition

Given a field $K$, a vector space $V$ over $K$ and a linear map $T:V \longrightarrow V$ of finite dimension with minimal polynomial $\mu(x)\in K[X]$, prove using the primary decomposition of V (...
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Confusion about the primary decomposition theorem in linear algebra

I am currently studying the Jordan canonical form which uses the primary decomposition. I have seen the generalised eigenspace decomposition and I know that the algebraic multiplicity which appears in ...
Wintermelon423's user avatar
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Proving that if $J=\bigcap_{n\geq0}I^n$ then $IJ = J$

I'm having some trouble with the following exercise: Let $R$ be a Noetherian commutative ring, $I$ and ideal and $J=\bigcap_{n\geq0}I^n$. Show that $IJ = J$. (Hint: Assume that $J\not\subseteq IJ$ ...
Eduardo Magalhães's user avatar
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Is there an algorithm like primary decomposition of an ideal that also assumes there are no zero divisors?

I have a large system of polynomial equations and am trying to construct an irreducible decomposition of the corresponding variety. It's large enough that feeding it to the standard radical/primary ...
Brent Baccala's user avatar
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Given an ideal $I$, when is $I^nR_P=P^sR_P$ for some $n,s \ge 1$ and some prime ideal $P$

Let $I$ be an ideal of a commutative Noetherian ring $R$. Under what hypotheses can we say that $I^nR_P=P^sR_P$ for some $n,s\ge 1$ and some prime ideal $P$ of $R$? Note that such a prime ideal $P$ ...
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Example of a nonzero ideal which is contained in every other non zero ideal.

I am searching for a ring in which $0$ ideal is irreducible but not primary. I have an idea to show that $0$ ideal is irreducible : We have to construct a ring $R$ in such a way that there exist a ...
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Definition of Primary Decomposition

When encountered with the word 'decomposition', we immediately think of a structure being broken down into and written as a (kind of) 'union' of simpler parts(say for example the decomposition of a ...
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A principal ideal is prime in a coprime localization -- is it prime in the ring?

Let $R$ be a normal Noetherian domain. Suppose there is $x \in R$ such that the principal ideal $xR$ has only one associated minimal prime ideal $\mathfrak{p}$. Assume there exists a multiplicative ...
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Is every subspace can be represented as T-cyclic for some T? [closed]

Definition of $T$-cyclic subspace: Let $T$ be an linear operator on $V$. Take $v \in V$. The subspace generated by $\text{span}(\{v,T(v),T^2(v),\cdots\})$ is called $T$-cyclic subspace generated by $v$...
Ganesh Karthi's user avatar
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Kernels of pairwise relatively prime polynomials polynomials

I'm trying to find a reference for the following theorem in linear algebra. Theorem. Let $k$ be a field, let $\mathfrak{P} \subseteq k[x]$ be a set of representatives of the irreducible polynomials in ...
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Find the primary decomposition of ideals in $\mathbb{Z}[\sqrt{5}]$ [duplicate]

I would like to find the minimal primary decompositions of, say, $(4)$ and $(1+\sqrt5)$ in $\mathbb{Z}[\sqrt{5}]$. In Find minimal primary decompositions of ideals there is a similar example, what ...
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Is this space Noetherian?

Define $R:=\mathbb{C}[x_0,x_1,\cdots]/(x_0x_1,x_1x_2,\cdots,x_{2n}x_{2n+1},\cdots)$ . R is a quotient ring of the ring of polynomials over complex field in inifitete variable. I found this ring in ...
George's user avatar
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On local rings $(R, \mathfrak m)$ such that $\text{Spec}(R)$ is disjoint union of $\text{Ass}(R)$ and $ \{\mathfrak m \}$

Let $(R, \mathfrak m)$ be a Noetherian local ring such that $\mathfrak m \notin \text{Ass}(R)$ and $\text{Spec}(R)=\text{Ass}(R)\cup \{\mathfrak m \}$. Then, must $R$ be Cohen-Macaulay? Of course the ...
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Irredundant primary decomposition of radical from irredundant primary decomposition under certain conditions

Let $R$ be a noetherian ring, and $\mathfrak{a}$ a proper ideal of $R$. Let all the associated primes of $R/\mathfrak{a}$ ($Ass\left(R/\mathfrak{a}\right)=\{\mathfrak{p}_{1},\dots,\mathfrak{p}_{r}\}$) ...
rubikman23's user avatar
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Proof that ideals are irreducible

In Hassett's book Introduction to Algebraic Geometry he states: $$ (x^2, xy, y^2 ) = (y+x, x^2) \cap (x,(y+x)^2)$$ and both ideals are irreducible (i.e., not writable as intersections of non-trivial ...
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Check on understanding Primary Decomposition Theorem

I've learned the Primary decomposition theorem recently and wanted to check if I have this right. Is this example correctly applied? Let $V$ be finite dim. and $T$ a linear map on $V$. Suppose $m_T(x)=...
jet's user avatar
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Decomposition of $(3)\subseteq \mathcal O_K$. Minimal polynomial of $\sqrt[3]{19}$ which gives the integral basis.

Let $\alpha=\sqrt[3]{19}$, $K=\mathbb Q(\alpha)$ and ring of integer is given $\mathcal O_K$. Now I want to calculate some prime decomposition. However, in Marcus Number Fields' notes, since extension ...
Micheal Brain Hurts's user avatar
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If $\text{Ass}(R/P^n)=\{P\}$ hold for infinitely many $n$ , then does it hold eventually for all large enough $n$?

Let $P$ be a prime ideal of a commutative Noetherian ring $R$. If $\text{Ass}(R/P^n)=\{P\}$ for infinitely many positive integers $n$, then is it true that $\text{Ass}(R/P^n)=\{P\}$ for all large ...
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Any ideal as an intersection of ideals primary to maximal ones

The Nullstellensatz says that any ideal $I \subset \mathbb{C}[x_1,\dots,x_n]$ has the property that $\sqrt{I} = \bigcap_{\text{maximal } \mathfrak{m} \supset I} \mathfrak{m}$ Is it also true that we ...
Sasha's user avatar
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1 answer
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The primary components of a space relative to a linear operator.

I am trying to understand a proof from N. Jacobson book ``Lectures in Abstract Algebra, II. Linear Algebra'', Chapter IV.8 (p.130, but I've attached the screen for convinience). Let $V$ be a finite-...
Butters Stotch's user avatar
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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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1 answer
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A question regarding primary decomposition of ideals

This question was left as an assignment in my class of commutative algebra but I was not able to completely solve it. Prove that $I = (x^2, xy)= (x) \cap (x^2,y) = (x) \cap (x^2,xy,y^n)$ for any $n \...
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1 vote
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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 ...
Diego Trujillo's user avatar
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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 ...
sagnik chakraborty's user avatar
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1 answer
226 views

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 $p(x)^n$ ...
Moisés Rodríguez's user avatar
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1 answer
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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,...
Jabberwocky's user avatar
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1 answer
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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, $i=1, \dots,s$....
MJane's user avatar
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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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1 answer
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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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1 answer
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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 ...
Dr. Scotti's user avatar
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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 ...
nekcihcyerewolf's user avatar
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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:...
Hola's user avatar
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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 ...
Snake Eyes's user avatar
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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 ...
Louis 's user avatar
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1 answer
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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 ...
defacto's user avatar
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2 answers
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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 ...
uno's user avatar
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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 ...
Conjecture's user avatar
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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^...
Math101's user avatar
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2 votes
1 answer
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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+...+...
autodidacti's user avatar
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102 views

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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2 answers
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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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