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Questions tagged [ideals]

An ideal is a subset of ring such that it is possible to make a quotient ring with respect to this subset. This is the most frequent use of the name ideal, but it is used in other areas of mathematics too: ideals in set theory and order theory (which are closely related), ideals in semigroups, ideals in Lie algebras.

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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 an primary decomposition of $(x^2, xy, xz, yz)$. I have read about a strategy of finding primary decomposition ...
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Proposition $(S_1,\ldots,S_r)$ from commutative algebra

I'm reading Qing Liu's Algebraic Geometry and Arithmetic Curves. I don't know what $(S_1,\ldots,S_r)$ in the following proposition(p. 30) exactly means. Proposition 1.11. Let $k$ be a field, ...
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Product of two coprime ideals in a commutative ring.

If $I$ and $J$ are two coprime ideals in a commutative ring $R$, i.e. $I+J=R$, then $IJ=I\cap J$. The above fact has been stated without proof in almost every textbook I have referred. No one seems ...
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If $P$ is a prime ideal of $R$ and $X\subseteq P$ then there's a minimal ideal [duplicate]

Let a ring $R$ and let $X\subseteq P\subset R$ such that $P$ is a prime ideal of $R$. Prove that there's a minimal prime ideal $M$ of $R$ such that $X\subseteq M\subseteq P$. I think of using Zorn's ...
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How to plot ideals of rings

Im trying to better understand ideals of rings and I think being able to visualize what I'm working with would help. I want to graph them (I'm talking mostly about quadratic rings), but I don't know ...
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1answer
55 views

Ideal with no zero divisors implies integral domain?

I'm trying to figure out a solution to the question: If a commutative ring $R$ has a nontrivial proper ideal $I$ that contains no nontrivial zero divisor of $R$, is $R$ an integral domain? I ...
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Surjections of polynomial rings, Krull dimension, and regular sequences

Let $k$ be a commutative ring with unity (I will ultimately want $k = \mathbb Z$) and $$\varphi\colon A = k[x_1,\ldots,x_n] \to k[y_1,\ldots,y_p] = B$$ a surjective ring homomorphism. (I am able to ...
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Factorization in $\mathbb{Q}[\sqrt{-5}]$ [closed]

Let $R$ be the ring of integers of $\mathbb{Q}[\sqrt{-5}]$. How can I write, say, $5R$ as the product of prime ideals in $R$?
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45 views

Find a maximal ideal in $F_7[x]/(x^2+5)$

$F_p=Z/(pZ)$ I'm stuck at this problem. I tried to find the irreducible factorisation of $x^2+5$, but that didn't help too much. Any ideas? and also could anyone explain the general philosophy with ...
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26 views

Let I be a f.g. ideal of R. Let S be the set of ideals J which do not include I. Show S has maximal element.

Let I be a f.g. ideal of R. Let S be the set of ideals J which do not include I. Show S has maximal element. My attempt is to construct a maximal element by taking the union of all elements in the ...
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19 views

Commutative ring with finitely many minimal primes [duplicate]

$A$ be a commutative ring with $1$ with finitely many minimal primes $\{p_1,\ldots,p_n\}.$ Then how can I show that $S^{-1}A \cong A_{p_1} \times\cdots \times A_{p_n},$ where $S=A \setminus \bigcup_{i=...
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Form of ideal generated by a set in non-commutative case

I am unable to visualize how a the ideal generated by a set looks like in case of non-commutative ring. Question: Suppose $R$ is a non-commutative ring with 1$\ne$0 and $R[x,y,z,w]$ be the polynomial ...
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1answer
21 views

Find all the ideals of $\mathbb { Z } [ i ]$ containing $(5)$

Problem : Find all the ideals of $\mathbb { Z } [ i ]$ containing $(5)$, the principal ideal generated by $5$. I already know that $\mathbb { Z } [ i ]$ is isomorphic to $\mathbb{ Z }[x]/(x^2+1)$ ...
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1answer
39 views

Finding associated primes of quotient modules

Consider the ideal $I=(a)\cap(a,b)^2=(a^2,ab)$ in $k[a,b]$ and set $R=k[a,b]/I$. The problem is to show that for all $n\ge 1$ and all $\lambda\ne 0$, the ideals $(b^n)$ and $(a+\lambda b^n)$ of $R$ ...
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Proof of every non-unit belongs to some maximal ideal. [duplicate]

I want to prove that every non-unit belongs to some maximal ideal. I did the following. Consider a commutative ring $R$ with unity. Consider a maximal ideal $M$. Also consider an element $r\notin M$. ...
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26 views

calculate intersection of two polynomial ideals

Show that in $\mathbb{C}[X,Y]$, the ideals $(X^3-X^2,X^2Y-X^2, XY-Y, Y^2-Y)$ and $(X^2,Y)\cap (X-1, Y-1)$ coincide. Is this a radical ideal? I can show that $(X^3-X^2,X^2Y-X^2, XY-X, Y^2-Y) \subseteq (...
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1answer
28 views

Equivalent definitions of primary ideals [duplicate]

Here are two definitions of a primary ideal. An ideal $I\subset A$ is primary if $I\ne A$ and $xy\in I\implies$ either $x\in I$ or $y^n\in I$ for some $n> 0$. An ideal $I\subset A$ is primary if $...
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1answer
27 views

Homogeneous elements of an ideal over a quotient ring

Let $k$ be a field. Consider the ideals $I_1=(x),I_2=(y),J=(x^2,y)$ of $R=k[x,y]/(xy,y^2)$. Show that the homogeneous elements of $J$ are contained in $I_1\cup I_2$ but that $J\not\subset I_1$ and $J\...
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Using the `initialIdeal` function in Macaulay2

My understanding is that there's an initialIdeal function in Macaulay2 for computing intial ideals with respect to a grading, specifically in the ...
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1answer
37 views

Ideals in Laurent polynomials over a field

Be $F$ a field and let $I$ be any ideal in $F[X,X^{-1}]$. For any $f = \sum_{n\in \mathbb{Z}}a_nX^n \in F[X,X^{-1}]$, define deg$^-(f) := \min\{n \in \mathbb{Z} \mid a_n \neq 0\}$. Consider the set $\...
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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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Finding inverses in quotient rings

In $ A=\mathbb{Z}[i]=\{a+bi \ : \ a,b \in \mathbb{Z}\} $ we consider $a=7+56i; \ b=3+3i; \ c=1+8i$. We will write $(a)$ to refer to the ideal generated by $a$ Find out whether the elements $\...
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1answer
50 views

Let $k$ be a non-algebraically closed field and $I\subset k[x_1,\dots, x_n]$ be maximal ideal. Is $V_{\bar{k}}(I)$ necessarily finite? [closed]

Let $k$ be a non-algebraically closed field and $I\subset k[x_1,\dots, x_n]$ be a maximal ideal. $\textbf{Q:}$ Is $V_{\bar{k}}(I)=\{x\in\bar{k}^n\vert \forall f\in I, f(x)=0\}$ necessarily finite?
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Product of ideals in $\mathbb{Z}[X]$

Consider the ideals $I = (2,X), J = (3,X) \in \mathbb{Z}[X]$. I want to show that the 'product set' $\Pi := \{ij \mid (i,j) \in I \times J\}$ is not an ideal in $\mathbb{Z}[X]$ and in particular, ...
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26 views

$\sigma$-ideal of subsets of $2^\omega$

I do not understand why here on the page 148 $\cal M^*_{2, K}$ is a $\sigma-$ideal of subsets of $2^\omega$. I even do not know the weaker statement: why it is an ideal?
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$R$ commutative ring with 1 and not every ideal is principal. Prove $R$ has ideal that is not principal.

I am wondering how to go about proving this, Let $R$ be a commutative ring with identity such that not every ideal of $R$ is principal. A) Use Zorn's lemma to show that $R$ has an ideal $J$ such ...
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1answer
29 views

Confused about quotient ring

For $I$ an ideal of a commutative ring $R$, I'm confused about the object $R/I$. My understanding is that this is defined by $R/I := \{rI: r \in R\} = \{\{ri: i \in I\}: r \in R\}$. However, by the ...
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44 views

Show that every non-zero ideal of the ring of rationals with odd denominator is generated by $2^{n}$ [duplicate]

Let $R \subset \mathbb{Q}$ be the subring $\left\{\frac{a}{b} \mid a, b \in \mathbb{Z}, b \text { odd }\right\}$. Prove that the ideals of $R$ are the zero ideal $\{0\}$ and $2^{n} R$ for $n \geq 0$. ...
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Proving $y-x^2, z-xy$ generate the ideal of the twisted cubic

This question comes from Hartshorne's exercise 1.2. He defines $Y:=\{(t,t^2,t^3)\in \mathbb{A}^3\mid t\in k\}$ and asks us, among other things, to find generators for the ideal $I(Y):=\{f\in k[x,y,z]\...
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Abstract Algebra: Quotient ring, ideal, and isomorphism

I need help with the following exam exercise, my teacher didn’t post the answer and I can’t manage to solve it. In $ A=\mathbb{Z}[i]=\{a+bi \ : \ a,b \in \mathbb{Z}\} $ we consider $a=7+56i; \ b=3+...
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Is this an example of comaximal ideals $I,J$ such that $IJ\not=I\cap J$?

Let $R$ denote the set of all finite formal sums of elements in the free group $\left<a,b\right>$ with the relation $a+b=1.$ Let $I$ be the principle ideal generated by $a$ and $J$ the principle ...
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$M_{i+1}/M_i\simeq R/P_i$ for some prime ideal $P_i$

Proposition 3.7 in Eisenbud says that for a f.g. module $M$ over a Noetherian ring there is a chain $$0=M_0\subset M_1\subset\dots\subset M_n=M$$ with $M_{i+1}/M_i\simeq R/P_i$ for some prime ideal $...
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2answers
35 views

Does a ring homomorphism $\phi: R \rightarrow S$ give rise to any map $\psi: R/I \rightarrow S/J?$

Let $R, S$ be rings. Suppose $\phi: R \rightarrow S$ is a ring homomorphism. Clearly we have a map $R \rightarrow S/J$ defined by the canonical map. However, for any ideal $I \subset R,$ can I ...
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There exists a prime ideal not containing non-nilpotent element [duplicate]

I've been trying to solve the following problem. Let $A$ be a commutative ring with identity and $a \in A$ a non-nilpotent element, i.e., $a^m \neq 0$ for all $m \in \mathbb{Z}^+$. Prove there exists ...
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1answer
20 views

Localization of the polynomial ring at a prime ideal modulo maximal ideal is isomorphic to polynomial ring modulo prime ideal.

Let $p \in K[T]$ irreducible, s.t. $\text{LC}(p) = 1$. Then $$ K[T]/(p) \cong K[T]_{(p)}/pK[T]_{(p)}.$$ What I have is: \begin{align*} &K[T] \hookrightarrow K[T]_{(p)} \text{ and } K[T]_{(p)} \...
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1answer
36 views

If $R$ is an n-fir, why free $R$-modules of rank at most $n$ have unique rank?

I am studying the Cohn's book "Free ideal Rings and Localization in General Rings", and there is something he takes for granted and I cannot find its reason in any part of the book. The question is: ...
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51 views

Is $ \{ (\alpha ^ 2, \alpha) | \alpha \in \mathbb{R}\} \subseteq \mathbb{C}^2 $ an algebraic variety?

Is $S = \{ (\alpha ^ 2, \alpha) | \alpha \in \mathbb{R}\} \subseteq \mathbb{C}^2 $ an algebraic variety? I think the answer is no, and this was my approach: If $S$ is an algebraic variety then $I(S)$...
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1answer
52 views

How to find an ideal of a ring

I'm fairly new to ideals in my algebra course, and I understand the basics of ideals, such that I is an ideal iff (I,+) is a subgroup of (R,+) (using normal subgroup tests) and for all $r \in R$ and $...
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1answer
18 views

Find an element that generates the Ideal

Given the ideal: $A$:={$\sum_{i∈I}T^i|$ $I $ finite,$|I|$ even} ⊂ $\mathbb{F}_2[X]$ find an element that generates it. Now I know that it has to be $A=a$ $\mathbb{F}_2[X]$, where $a$ is the element ...
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1answer
61 views

Units in a discrete valuation ring.

I'm doing problem 2.26 in the book "algebraic curves" by Fulton: http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf One is given two DVR's R and S both of which have the same field of fractions $K$...
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How do you prove there is a bijection between an isomorphism and a set of orthogonal idempotents?

Let $A$ be a commutative ring with unity. How do you prove there is a bijection between: An isomorphism as a product of rings $\phi: A \longrightarrow A_1\times\cdots\times A_n$ A decomposition as ...
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1answer
82 views

Why is $6\mathbb{Z} + 9\mathbb{Z}$ a principal ideal of $\mathbb{Z}$?

Ler $R$ denote a commutative ring with identity. For $R = \mathbb{Z}$, why is the ideal $6\mathbb{Z} + 9\mathbb{Z}$ principal? Any help where to start would be appreciated.
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1answer
51 views

Finding all ideals $T$ in number ring $\mathbb Z[\sqrt{-3}]$ s.t. $\langle 4 \rangle \subset T$.

I want to find all ideals $T$ in number ring $\mathbb Z[\sqrt{-3}]$ s.t. $\langle 4 \rangle \subset T$. My idea: $ 4= 2\times2 = (1-\sqrt{-3}) (1+\sqrt{-3})$ so $\langle4\rangle = \langle 2\rangle \...
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3answers
46 views

Is this ideal principal?

Let $I = \{ f(x)∈ \mathbb{Z}[x] | f(0) \text{ is an even integer}\}$ This is an ideal of the ring $R = \mathbb{Z}[x]$. Is it principal? I have the definition of a principal ideal but I'm unsure if ...
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1answer
31 views

Product of the two subrings

These are problems bothering me. Let $S$ and $T$ are subrings of a ring $R$. And $I$ and $J$ are ideals of the $R$. Question 1 Let product of subrings like ideal product $IJ$ which means $ST = \{...
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4answers
97 views

How to prove this isomorphism of a quotient ring

I'm trying to understand more about ring theory and the concept of ideals has been confusing. I'm trying to understand why this is true: $\mathbb Z[\sqrt{-5}]/(1+\sqrt{-5})\simeq\mathbb Z/6\mathbb ...
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2answers
67 views

Ideal of $\mathbb{Z}[x]$ generated by $3$ and $x^2+1$ is proper

Let $I$ be the ideal of $\mathbb{Z}[x]$ generated by $3$ and $x^2+1$. Show that 1) $I$ is proper, 2) $\phi_a(I)=\mathbb{Z}$ for all $a\in\mathbb{Z}$, where $\phi_a(I):=\{f(a)\mid f\in I\}$ (...
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3answers
67 views

Ideal generated by two elements is maximal in $\mathbb{Z}[x]$

The question is to show that $I = (x^4 + x^3 + x^2 + x + 1, 2) \subset \mathbb{Z}[x]$ is a maximal ideal. I'm familiar with the results that $R / I$ is a field iff $I$ is maximal, and $R/I$ is a ...
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0answers
48 views

How to prove that $\dim_k(k[x_1,\dotsc,x_n]/\sqrt{I})=|V(I)|$?

Let $k$ be an algebraically closed field. How to prove that $\dim_k(k[x_1,\dotsc,x_n]/\sqrt{I})=|V(I)|$? I know that Hilbert Nullstellensatz will be required, but I can't figure out how. With the ...
2
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1answer
28 views

Find a homomorphism that maps a point in a Boolean algebra into the image of a proper filter

Let $\mathbb B$ be a Boolean algebra. Let $F$ be a proper filter in $\mathbb B$ (i.e. $0 \notin F$), and let $I$ be its dual ideal. Suppose there exists $a \in \mathbb B$ such that $a \notin F \cup I$....