# 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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### $\text{depth }A\leq \text{depth }_{\frak{p}}A + \dim(A/{\frak{p}})$, 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 ...
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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 $m \mapsto x m$. ...
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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 ...
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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 $J\not\subseteq IJ$ ...
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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 ...
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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 $v$...
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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 ...
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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 $R/\mathfrak{a}$ ($Ass\left(R/\mathfrak{a}\right)=\{\mathfrak{p}_{1},\dots,\mathfrak{p}_{r}\}$) ...
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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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### 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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### 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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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 ...