# Questions tagged [primary-decomposition]

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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,...
1 vote
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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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### 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 ...
1 vote
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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 ...
1 vote
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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 ...
1 vote
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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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### 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 ...
1 vote
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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 ...
1 vote
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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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### 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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### 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 ...