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Questions tagged [primary-decomposition]

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Hoffman and Kunze ,Linear algebra Sec 7.4 exercise 4

Construct a linear operator $T$ with minimal polynomial $ x^2(x-1)^2 $ and characteristic polynomial $x^3(x-1)^4$. Describe the primary decomposition of the vector space under $T$ and find the ...
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On two kinds of powers of Ideals related to Symbolic powers

All rings below are commutative, with unity and Noetherian. For a prime ideal $P$ in $R$, if $\phi_P : R \to R_P$ is the localisation map, and $I$ is an ideal in $R_P$, let us write $I \cap R$ to ...
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Analogue of Krull intersection theorem for Symbolic powers of ideals

All rings below are commutative with unity and Noetherian. Let $R$ be a domain or a local ring and $J$ be a proper ideal. Is it true that $\bigcap_{n>1} J^{(n)}=(0)$ ? If this is not true under ...
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Is there a way to extract a primary decomposition of $(0)\subset R/I$ given a decomposition of $I\subset R$?

Suppose we have a (minimal) primary decomposition of an ideal $I$ in a polynomial ring $R$ over a field; for simplicity assume $I=I_1\cap I_2$ where $I_j$ is $P_j$-primary. Is there a way to extract ...
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Primary decomposition of powers of a monomial (edge) ideal in a three variable polynomial ring

Consider the ideal $J=(xy,yz,zx)$ in $R=\mathbb C[x,y,z]$. As seen here Associated primes of the square of a monomial ideal , $Ass_R (R/J^n)=\{(x,y); (y,z);(z,x);(x,y,z)\}, \forall n \ge 2$. My ...
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Primary decomposition of $(x^2, xy, xz, yz)$ in $K[x, y, z]$

Suppose $K$ is a field and consider the ideal $(x^2, xy, xz, yz)$ of $K[x, y, z]$. Find a primary decomposition of $(x^2, xy, xz, yz)$. I have read about a strategy of finding primary decomposition ...
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Associated primes of the square of a monomial ideal

Consider the ideal $J=(xy,yz,zx)$ in $R=\mathbb C[x,y,z]$. How to show that $(x,y,z) \in \mathrm{Ass}_R (R/J^2)$ ?
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On finiteness of $\cup_{n\ge 1 } \operatorname{Ass}_R (R/I^n)$

Let $I$ be an ideal of a commutative noetherian ring $R$. How to prove that $\bigcup_{n\ge 1 }\operatorname{Ass}_R (R/I^n)$ is finite ? I am aware of Brodmann's result about Asymptotic stability ...
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On a subset of the associated primes of tensor product of modules

For a module $M$ over a commutative ring $R$, let $\operatorname{Ass}_R (M):=\{\operatorname{ann}_R (m) \mid m\in M$ and $\operatorname{ann}_R(m) \in \operatorname{Spec}(R)\}$. If $M,N$ are ...
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If $B$ is an $A$-algebra of Noetherian rings then $\text{Ass}_{A}M=\{f^{-1}(p):p \in \text{Ass}_{B}M\}$ for a f.g. $B$-module $M.$

Let $A,B$ Noetherian rings and $f: A \to B$ be a ring homomorphism. Let $M$ be a finitely generated $B$ module. Then want to show that $\text{Ass}_{A}M=\{f^{-1}(p):p \in \text{Ass}_{B}M\}.$ So far I ...
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Prove that $M_i/M'=\ker(M/M_i\to (M/M_i)_{P_i})$

As pointed out here, I believe, there is a typo in Eisenbud's Theorem 3.10(b). The modified statement is this: Let $M'$ be a proper submodule of a f.g. module $M$ over a Noetherian ring $R$ and let ...
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A typo in Eisenbud's Theorem 3.10?

Let $R$ be a Noetherian ring and $M$ a f.g. $R$-module. Let $M'$ be a proper submodule of $M$ and let $M'=\cap_{i=1}^n M_i$ be a primary decomposition with $M_i$ a $P_i$-primary submodule. Part (c) ...
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The notion of $P$-primary component

p.95 in Eisenbud: If $M$ is any module and $P$ is a minimal prime over $\operatorname{Ann}(M)$, then the submodule $M'\subset M$ defined by $$M'=\ker(M\to M_P)$$ is $P$-primary because $M/M'$ ...
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Intersection of ideals and primary ideals

In T.Y. Lam book Exercises in Modules and Rings here, page 84 Let $\mathbb{K}$ be a field, $R=\mathbb{K}[X,Y], \space \space $ $ I=(Y^2, XY)$ , $ Q_{1}=(Y) $ and $Q_{2}=(Y^2 , X + tY)$ ...
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Factorization in the three variable function

Is there any way to estimate the $f(\theta)$ in the following equation? $exp(\lambda_1 \times cos^2\theta) \times exp(\lambda_2 \times cos\theta) = f(\theta) \times exp( \lambda_1+\lambda_2) $ $\...
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Ideals with equal squares in a Noetherian UFD of dimension 2

Let $I$ and $J$ be two ideals in $\mathbb C[X,Y]$ such that $\mu(I)=\mu(J) \le 3$ and $I^2=J^2$ . Then is it necessarily true that $I=J$ ? If not, then is it at least true if we assume $I,J$ are ...
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Primary ideals are those whose quotient has irreducible prime spectrum. [duplicate]

I'm currently trying to figure out primary ideals and I think I proved the following geometric statement, but I'm not sure if I did it right and would like to have some feedback. Lemma: An ideal $\...
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Writing a projective scheme as a union of irreducible subschemes.

Suppose $i: X \hookrightarrow \mathbb{P}^n_k$ is a projective scheme, over some field $k$ (I'm not sure if algebraically closed is necessary here). So $X$ comes together with an ideal sheaf and a ...
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Embedded primes in a finitely generated algebra over an algebraically closed field

Let $F$ be an algebraically closed field. Consider the polynomial ring $F[X,Y,Z]$ and its quotient by the ideal $J=(X^aZ, XY^b)$. I want to find all the $(a,b)$, for $a,b$ positive integers, such that ...
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Primary decomposition of $J = (XZ-Y^2, X^3-YZ)$

In occassions where I need to find the minimal primes associated to an ideal / find a primary decomp., sometimes I can do it just fine, sometimes I find myself completely blind in the search for it. ...
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Length of a module is sum of length of modules localization at each associated prime.

Let $M$ be an $A$-module with finite length, i.e., $\ell_{A}(M)< \infty .$ Also $A$ is a Noetherian ring. Then I want to show that $$\ell_{A}(M)= \sum_{p \in \text{Ass}(M)} \ell_{A_p}(M_p).$$ I ...
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Artin-Rees property in commutative Noetherian rings with unit

I am trying to prove that if $R$ is a commutative Northerian ring with $1$, for all ideals $I$, $E$, then $E \cap I^n \subseteq EI$ for some $n\in \mathbb{N}$. I have tried to go in this direction, ...
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On cancelling $\mathfrak m$-primary ideal of regular local ring $(R,\mathfrak m)$

Let $(R,\mathfrak m)$ be a regular local ring (https://en.wikipedia.org/wiki/Regular_local_ring) . Let $J$ be an $\mathfrak m$-primary ideal such that $J^2=\mathfrak mJ$. Then is it true that $J=\...
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A characterization of regular local ring using primary ideal?

Let $(R,\mathfrak m)$ be a Noetherian local domain such that for every $\mathfrak m$-primary ideal $I$ with $I^2\subsetneq \mathfrak m^2$, we have $I^2\ne \mathfrak mI$. Then is it true that $R$ is ...
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If $W$ is T-invariant, $W=(W\cap W_{1})\oplus\cdots\oplus (W\cap W_{k})$

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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Find inner product that meets primary decomposition criterias

Let $W_1, W_2$ be subspaces over $\mathbb C$ and let $V = W_1 \oplus W_2$ Assume $\langle\cdot{,}\cdot\rangle_1$ is the inner product on $W_1$ and $\langle\cdot{,}\cdot\rangle_2$ is the inner product ...
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Primary decomposition of $I=(x_1)\cdot(x_1,x_2)\cdot\cdot\cdot(x_1,..,x_n)$

Let $S=k[x_1,...,x_n]$ the polynomial ring in $n$ indeterminates. Let $I=(x_1)\cdot(x_1,x_2)\cdot\cdot\cdot(x_1,..,x_n)$ this monomial ideal and I have to find an irredundant primary decomposition for ...
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Decomposition in irreducible factors in a factor ring

I'm currently learning for my Commutative Algebra exam and I have the following question: Given $\mathbb Z$ the ring of all integers and the ring $\mathbb Z[X, Y]$ the ring of polynomials in ...
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How to find the cyclic vectors when finding the Rational form of a given matrix?

For a given matrix, say $A$, of dimension $n$, in $\mathbb{F}^{n\times n}$, the characteristic polynomial of $A$ is of the form $$c_A = p_1^{d_1}*...*p_k^{d_k},$$ and the minimal polynomial of the ...
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A confustion about the proof of Primary Decomposition theorem

In proofWiki, in the proof of Primary Decomposition Theorem while showing that (in the induction step) $$\ker \left({p_i \left({T \restriction_W}\right)^{a_i} }\right) = \ker \left({p_i \left({T}\...