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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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Quotient cancellation for invertible ideals of orders in quadratic fields

Let $K/\mathbb{Q}$ be an imaginary quadratic field, $m\ge 1$ be a positive integer and let $\mathcal{O}=\mathbb{Z}+m\mathcal{O}_K\subset \mathcal{O}_K$ be the unique order of index (equivalently, ...
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When the element-wise product of two ideals produces an ideal

Consider the ring $R=\mathbb C[X,Y]$. For every two ideals $I,J$ of $R$, define $I*J:=\{ij : i\in I, j\in J\}$. Now definitely, $I*J=J*I$ always holds. If $I$ is principal, then actually $I*J$ is an ...
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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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cancellation ideals in Noetherian domain

For an ideal $I$ in a commutative ring with unity $R$, let us say $I$ is a cancellation ideal if for ideals $J,K$, $IJ \cong IK \implies J \cong K$. Now let $I$ be a cancellation ideal in a ...
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The radical of the product of two ideals is the intersection of the radicals of the ideals

Let $R$ be a Noetherian ring (not sure if "Noetherianness" is needed). I'm trying to show that $r(IJ)=r(I)\cap r(J)$ ($r$ for radical). One direction looks easy. If $x\in r(IJ)$, then for some $k$, $...
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On ideal quotient of intersection of fractional ideals

Let $R$ be an integral domain with fraction field $K$. For a fractional ideal $I$ (i.e. $I$ is an $R$-submodule of $K$ such that $\exists 0\ne r\in R $ with $rI \subseteq R$) of $R$, define $(R:I)=\{x\...
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There exists an element $x$ of $M$ such that $W^{-1}R\dfrac{x}{1}=W^{-1}Rz$ and $\operatorname{ann}_R x=\operatorname{ann}_{W^{-1}R} z\cap R$

Let $R$ be a Noetherian ring, $W$ a multiplicative subset of $R$, $M$ an $R$-module, and $z$ an element of $W^{-1}M$. Show that there exists an element $x$ of $M$ so that $W^{-1}R\dfrac{x}{1}=W^{-1}Rz$...
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Is it true that the generators of a commutative ring is finite?

So I am thinking about the structure of a commutative ring $R$. Give a family of principal ideals $A=\{(x_{i})\}$, and if $R=\bigcup_{i}(x_{i})$, is it true that there are finite number of principal ...
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On integral closedness of formal power series ring over an integrally closed domain satisfying Krull intersection principle

Let $R$ be a normal domain (i.e. an integral domain integrally closed in its fraction field) such that for every non-unit $t\in R$, $\cap_{n\ge 1} (t^n)=(0)$ ; then is it true that $R[[X]]$ is normal (...
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$I$ an irreducible ideal in a PID is a prime ideal

$I$ is irreducible ideal of $R$ when $R$ is a PID , I want to show it's a prime ideal. The definition I am familiar with: $I$ a proper ideal of $R$ is an irreducible ideal if for any two ideals $J,K$ ...
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Inclusion preserving map between ideal spaces is continuous in the Fell topology

Let $A$ be a $C^*$-algebras with ideal space $\mathcal{I} (A)$ and equip $\mathcal{I} (A)$ with the Fell topology, i.e. the topology generated by the subbase $U_{I}:=\left\{ J\in{\cal I}\left(A\right)\...
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ideal extension in polynomial ring over quotient field

This is probably easy, but I can't seem to figure it out. I have a $K$-algebra $B$, which is a domain and an irreducible polynomial $f \in K[x]$. Why does $B[x]/f$ embed in $Q(B)[x]/f$ where $Q(B)$ is ...
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If $\mathbb{Z}\ni \underbrace{z}_{\ne 0} \in I \trianglelefteq R:=\{a+\sqrt7b:a,b\in\mathbb{Z}\}$ then $[R:I] < \infty$

Let a ring $R:=\{a+\sqrt7b:a,b\in\mathbb{Z}\}$. Let an ideal $I\trianglelefteq R$ such that $z\in I$ for some $0\ne z\in\mathbb{Z}$. Prove that $[R:I]<\infty$. I showed that $z\in I$ for some $z&...
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Algebras finite-dimensional over their base field are Artinian.

The standard proof that a $k$-algebra $A$ that is finite-dimensional as a $k$-vector space is Artinian goes as follows: "Suppose we have an infinite descending chain of ideals $I_1 \supseteq I_2 \...
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Representing an finite-dimensional algebra over a field using a quotient ring

My professor was talking about different ways to think about an algebra $A$ over a field $k$. One that she mentioned briefly but didn't go into much detail on was that of a quotient ring. Roughly the ...
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Representing an algebra over a field using a quotient ring

My professor was talking about different ways to think about an algebra $A$ over a field $k$. One that she mentioned briefly but didn't go into much detail on was that of a quotient ring. Roughly the ...
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Maximality of an ideal for showing that an algebra is in fact a field

I have an algebra $A$ over the field $F$, with the finite dimensionality $n$ as a vector space over $F$. I can also assume that $A$ is an integral domain. Assuming that $v_1,...,v_n$ is a spanning ...
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If a ring is noetherian, then so is its subring

Suppose $A\subset R$ are rings so that $R$ is also an $A$-module. Assume further that there is an $A$-module homomorphism $R\to A$ that is identity on $A$. How to deduce that $A$ is noetherian if $R$ ...
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Invertible elements of $\mathbb{Z}_3[x] / (x^4+x^3-1)^3$

Let $F=\mathbb{Z}/3\mathbb{Z}$, $h(x)=x^4+x^3-1$, $R = F[x]/(h(x)^3)$. I know $R$ has $4$ ideals and $1$ maximal ideal. Let $M$ be the maximal ideal $(h(x))/(h(x)^3)$ I need to find the number of ...
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Localization correspondence

This is taken from Neukirch's algebraic number theory Proposition 12.3. Proposition (12.3). If $a\neq 0$ is an ideal of an order (one dimensional Noetherian integral domain) $o$, then: $o/a = \...
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Proving $R-S$ contains a prime ideal when $S$ is a multiplicative set

I'm mainly trying to prove that If $0\not \in S\subseteq R$ is a multiplicative subset of a commutative ring $R$ with identity. Then $R-S$ contains a prime ideal. Now, by using Zorn's lemma, ...
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Passing from a set of generators to a set of homogeneous generators

Consider a graded ring $R=R_0\oplus R_1\oplus\dots$. Suppose the irrelevant ideal $R_1\oplus R_2\oplus \dots$ is finitely generated. How come we can assume that it is generated by a finite number of ...
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Prime ideals in $R[X]/I$

How many prime ideals do we have in $R[X]/I$ if $I =⟨(X^2 − 1)^5⟩$ ???
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Consider the order $\mathbb{Z}[\sqrt[4]{24}]$. Find all ideals of norm 100.

I have found that the ring of integers is $\mathbb{Z}[\alpha, \alpha^3/4]$ where $\alpha = \sqrt[4]{24}$. I also know that in the ring of integers $(5)$ factors as two ideals of norm $25$, and $(2)$ ...
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Ideals in an infinite direct product of fields

Let $R$ and $S$ be fields. Consider the ring $R \times S$. Since $R$ (resp. $S$) is a field, the ideals of $R$ (resp. $S$) are $\{0_R\}$ and $R$ (resp. $\{0_S\}$ and $S$). Since every ideal of the ...
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Finiteness of an algebraic variety $V(I) \subseteq \mathbb{C}^n$ where $I$ is a zero dimensional ideal of $\mathbb{C}[x_1,\dots,x_n]$

How to demonstrate this?: Let I be an ideal of $\mathbb{C}[x_1,\dots,x_n]$ such that $\frac{\mathbb{C}[x_1,\dots,x_n]}{I}$ has finite dimension ($I$ is a zero dimensional ideal). Then $V(I) \subseteq ...
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Is there a ring containing ideals where the union of the ideals forms a subring but not an ideal?

In general the union of two subrings is not a subring, and likewise for the union of two ideals. However, I was wondering if there exists an example of a union of ideals that is actually a subring but ...
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If two ideals agree on $K\lhd R$ and yield the same quotients $\pi_K(I)=\pi_K(J)$, are they equal?

Inspired by my lectures on the basics of commutative algebra, I tried to investigate whether the noetherian property still holds when taking extensions (here meaning: if $I\lhd R$ and $R/I$ both ...
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Does $\frac{b}{s} \in S^{-1} I$ directly implies that $b \in I$?

Let $R$ be a commutative ring with identity $1_R$, $0 \not \in S\subseteq R$ be a multiplicative set, and $I\subseteq R$ be an ideal of $R$.Consider the ring of quotients $S^{-1}I$. I was trying to ...
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C* algebra exact sequences and ideals

if you have C* algebras $A,B$ and $C$ and $\exists$ a short exact sequence as follows $0\rightarrow A\rightarrow B \rightarrow C \rightarrow 0 $ where the functions are $\phi$ and $\psi$ respectively, ...
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Is the ideal generated by ${4,x}$ a principal ideal in $Z[x]$? [duplicate]

I've : $I=<p,x>$ is not a principal ideal in $Z[x]$ where p is prime. My question : Is $I=<p,x>$ a principal ideal in $Z[x]$ where p is not a prime? More particularly, is the ideal ...
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Class Number Calculation of a Real Quadratic Number Field

I am looking at the example below. Can anyone explain how they end up with the contradiction. Why do they reduce $a^2-65b^2$ modulo $5$ to show that it has no integer solutions?
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Prove $R_{\mathfrak p}$ has only one maximal ideal $\mathfrak pR_{\mathfrak p}$

$\mathfrak p$ is prime ideal of commtative ring $R$. localization $R_{\mathfrak p}:=(R-{\mathfrak p})^{-1}R$. We know $\mathfrak pR_{\mathfrak p}=(R-{\mathfrak p})^{-1}\mathfrak p$ is prime ideal of $...
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Relation between $k[a,b]$ and $k[c,d]$, given that $(a,b)=(c,d)$.

Let $k$ be a field of characteristic zero, $R$ a commutative $k$-algebra, $a,b,c,d \in R$ such that the ideal generated by $a$ and $b$, $I_{a,b}=(a,b)$, equals the ideal generated by $c$ and $d$, $I_{...
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Showing that an ideal in $\mathbb{Z}[\sqrt{-21}]$ is principal

I have $\mathfrak{a}=(5,\sqrt{-21}-2).$ Can anyone tell me why $\mathfrak{a}^2$ is principal? I have multiplied the ideals out to obtain $$(5, 5\sqrt{-21}-10,-17-4\sqrt{-21}) $$ How does this reduce ...
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Showing that an ideal is not principal in $\mathbb{Z}[\sqrt{-21}]$

I am trying to show that the ideal $$(2,\sqrt{-21}-1)(3, \sqrt{-21}) $$ is not principal in $\mathbb{Z}[\sqrt{-21}]$. Can anyone help with this?
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If $I$ is a maximal ideal and $a\in R -I$, then the assumption that $I + (a) = R$ gives a contradiction

While I'm trying to prove that Let $S$ be a multiplicative set in the commutative ring $R$ with identity s.t $0 \not \in S$.Let $I$ be a maximal ideal in $S^c = R - S$. Then show that $I$ is a ...
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Prime ideal containing an ideal and not intersecting a multiplicatively closed set [closed]

Suppose $R$ is a ring and $I$ an ideal of $R$, moreover suppose $M$ is a multiplicatively closed subset of $R$ such that $I$ does not intersect $M$ can you always find a prime ideal $I\subset P$ also ...
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Gaussian Integers form an Euclidean Ring

A Ring $R$ is called euclidean if a map $f:R\backslash {0} \rightarrow \mathbb{N}$ exists with the following properties: For two elements $a,b \in R$ with $b\neq 0$ there exist $q,r\in R$ with: (i) $...
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1answer
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Ideal of matrix ring

I'm going over some exercises and I'm not quite sure if I completely understand this one. Let $R=M_3(\mathbb{Q})$, i.e. $R$ is the ring of all $3\times3$ matrices over rational numbers. Describe ...
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How to prove a polynomial in $\mathbb{Z}[x,y]$ is irreducible

I had the following question: Let $f(x,y) = y^5+xy^2+x \in \mathbb{Z}[x,y]$ and let: $I_2 = (f(x,y),x-1,2)$ and $I_3 = (f(x,y),x-1,3)$ be two ideals in $\mathbb{Z}[x,y]$. Prove that $f(x,...
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Fundamental Theorem on Homomorphisms - Application

I want to show 1.) For $n,m\in \mathbb Z$ with $m|n$ there exists a ring homomorphism between $\mathbb{Z}/n\mathbb{Z} \rightarrow \mathbb{Z}/m\mathbb{Z}$. I know that there is the canonical ...
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How to prove $\Bbb Z[i]/(1+2i)\cong \Bbb Z_5$? [duplicate]

How to prove $\Bbb Z[i]/(1+2i)\cong \Bbb Z_5$? My method is $$(1+2i)=\big\{a+bi丨a+2b≡0\pmod 5\big\},$$ So any $a+bi$ in $\Bbb Z(i)$,we got $$a+bi=(b-2a)i+a(1+2i).$$ So $\Bbb Z[i]/(1+2i)=\big\{0,[...
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Principal (and not principal) ideals of the ring $\mathbb{R}^{\mathbb{R}}$

Doing some exercises on rings and ideals I found some hurdles on showing that some ideals on the ring $\left(\mathbb{R}^{\mathbb{R}}, +, \cdot\right)$ are (or not) principal. Hints? Let be $f,g \...
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195 views

Proving that the radical of an ideal is an ideal [duplicate]

I'm trying to solve problem 1.18 in Fulton's 'Algebraic Curves', which is illustrated in the attached image, but I'm having some difficulties understanding the first part. The ring R is assumed to be ...
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1answer
41 views

Let $R$ be a ring and $X$ a non-empty set. Considering $A=R^X$, describe the ring $(A,+,\cdot)$

An exercise asks me to describe this ring: Let $R$ be a ring and $X$ a non-empty set. Consider $A=R^X$ with sum and multiplication defined by: $\forall \,\,f,g\in A, \forall x\in X$: $\,\,\,\,(f+g)(...
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45 views

a.c.c. and d.c.c. on radical ideals in commutative ring of dimension zero

Let $R$ be a commutative ring with unity of dimension zero (i.e. every prime ideal is maximal). Does any of the following two conditions imply the other : 1) $R$ satisfies a.c.c. on radical ideals ...
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61 views

Prove two Ideal are equal

Consider ideals of the ring $\frac{F[x]}{x^n-1}$ where $F$ is an arbitrary finite field. Suppose $g(x) | (x^n-1)$ and $f(x) | (x^n-1)$ . Also suppose $A = \left <g(x)\right>$ be the ideal that ...
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97 views

ring in which every proper ideal is primary

Let $R$ be a commutative ring in which every proper ideal is a primary ideal. Prove that $R$ has at most one nonzero prime ideal and a proper ideal $I$ is prime if and only if $[$ for all $x \in R$ if ...
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Commutative local ring with $10$ ideals

Let $R$ be a commutative ring with unity with exactly $10$ ideals (including $\{0\}$ and $R$ ) . Then is it true that $R$ is a Principal Ideal Ring ? My Work: I know that any commutative ring with $5$...