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, ...

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Identities about ideals

If $A,B \subseteq K^n$ show the following: If $A \subseteq B$, then $I(B) \subseteq I(A)$. $I(A \cup B)=I(A) \cap I(B)$ Could you give me a hint, how the above identities could be proven? EDIT: ...
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Ideal in the ring of quotients

I got some problems with the following: Let $R$ be a ring with $1\in R$. Let $S\subset R$ a subset which is closed under multiplication and contains $1$. On the set $R \times S$ we define an ...
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Localization of polynomial ring as differentiable functions

Let $a \in \mathbb{R}$ be a point and $S=\mathbb{R}[x]_{(x-a)}$ the localization of the polynomial ring $\mathbb{R}[x]$ with maximal ideal $(x-a)$. i) Describe the elements of $S$ as differentiable ...
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Rings and prime ideals

While doing some exercises about rings and prime ideals i got stuck with the following: Having a ring R: {$a + b \sqrt7 | a,b \in \mathbb{Z}$}, being a subring of $\mathbb{R}$, and knowing that ...
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Why is the naive notion of a product ideal not necessarily additively closed? [duplicate]

Considering the product ideal $IJ = \{ \sum_{i=1}^n a_ib_i | a_i \in I, b_i \in J \forall i\}$, I've always seen it written that the more naive notion $IJ = \{ ij | i \in I, j \in J\}$ is not an ideal ...
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Determining if (3) is a maximal ideal in $\mathbb{Z}[\sqrt{7}]$.

As far as I can tell, the tools I have for determining if an ideal I of a ring R is maximal is either: Determine another ideal it is contained within, or look at the quotient ring $R/I$ and determine ...
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Left ideals of matrix rings are direct sum of column spaces?

Let $\mathbb K$ be a field and $M_n(\mathbb K)$ be the ring of the $n\times n$ matrices with entries in $\mathbb K$. Let $C_j\subset M_n(\mathbb K)$ be the subspace of all matrices which have all ...
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Prove $\bigcup _{k=1}^n I_k$ is in ideal

Let $n$ ideals such that $I_1 \subset I_2 \subset \cdots \subset I_n$. Prove that $J=\bigcup _{k=1}^n I_k$ is also an ideal. We need to show three things: $0\in J$. Trivial... $a,b \in J \implies ...
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Prime ideals in $\mathbb{Q}[X]$

Could you tell me why prime ideals in $\mathbb{Q}[X]$ are of the form $(q(x))$ where $q \in \mathbb{Q}[X]$ is irreducible?
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Proving that an ideal is prime - is it correct?

I need to prove that although $X^2 + 3X +1 \in \mathbb{Z} [X]$ is irreducible, the ideals $(5,X^2 + 3X +1 )$ and $(11, X^2 + 3X +1)$ are not prime. I know that an ideal $I$ is prime iff ...
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ideal in the ring of smooth functions

What is an ideal $I$ of the ring of smooth functions $C^{\infty}(\mathbb R)$ which is not finitely generated and for all $x\in\mathbb R$ there exist $f\in I$ such as $ f(x)\neq 0$.
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Ideals-algebraic set

Notice that in $\mathbb{C}[X,Y,Z]$: $$V(Y-X^2,Z-X^3) = \{ (t,t^2,t^3) \mid t \in \mathbb{C}\}$$ In addition, show that: $$I(V(Y-X^2,Z-X^3)) = \langle Y-X^2,Z-X^3 \rangle$$ Finally, prove that the ...
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72 views

Should it stand that $\gcd(f(x), g(x))=1$?

If we have an ideal of the form $I=\langle f(x), g(x)\rangle\subseteq\Bbb Z[x] $ should it stand that $I=\langle \gcd(f(x),g(x))\rangle$? For example, if we have the ideal $I=\langle 2,x \rangle $ ...
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Prime ideal $P$ in $R$ coprime to the conductor plus the localization $R_{P}$ is a DVR implies that $P$ is invertible

I have the following situation: Let $B \subseteq B'$ be a ring extension such that $\text{Quot}(B) = \text{Quot}(B') =: K$ and $\text{dim}(B) = \text{dim}(B') = 1$ where $B'$ is a Dedekind domain. ...
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Vanishing Ideal

Let $$M:=\{(0,0,z), (-1/3,-1/3,z), (1/6(1+i\sqrt{3}),1/6(1-i\sqrt{3}),z), (1/6(1-i\sqrt{3}),1/6(1+i\sqrt{3}),z) |z\in \mathbb{C}\}.$$ I want to find the vanishing ideal $$I(M):=\{f \in ...
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Can someone explain me the sentence about ideals?

Can someone explain me the sentence: "If $R=K[x]$ the prime ideals are $\langle f(x)\rangle $ where $f(x)$ is an irreducible polynomial in $K[x]$ and $\langle 0\rangle $, and again $\langle ...
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63 views

Ideal is contained in a union of prime ideal

Let $I \subset R$ be an Ideal and $P_i$ $(i=\{1,...,n\})$ prime Ideals with $I\subseteq\bigcup_{i=1}^nP_i$ Prove that then $I$ is contained in one $P_i$. I don't know how to show this because I ...
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Is $\left\{ f\in F[x] | \deg(f) < m \right\} \cup \left\{0\right\}$ an ideal?

Is $\left\{ f\in F[x] | \deg(f) < m \right\} \cup \left\{0\right\}$ an ideal? $0$ is clearly in the set Easy to see that $a,b\in I \implies a+b\in I$ The last demand is that $a\in I, r\in R ...
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Prove that $I$ is a maximal ideal

I have a question. To show that the ideal $I=\langle f(x)\rangle $ is a maximal ideal of $K[x]$ do I have to show that $f(x)$ is irreducible in $K[x]$? Or is there an other way to prove that $I$ is a ...
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Norm of Prime Ideal

Show that the norm of a prime ideal in a number field $K$ is a power of some prime number, i.e., if $P$ is a prime ideal in $O_K$ for some number field $K$, then $N_\mathbb{Q}^K(P)=p^n$ for some ...
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233 views

Prove that $(x^3-2)$ is a maximal ideal of $\Bbb Q[x]$

Prove $(x^3-2)$ is maximal ideal of $\Bbb Q[x]$ using isomorphism theorems for rings. I tried using the second isomorphism theorem for rings, to use that $( x ^ 3-2)$ is maximal if and only if ...
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Prove that $I_1\cap I_2$ cannot be a prime ideal.

Prove the following: Let $I_1$ and $I_2$ be ideals of a ring $R$ such that $I_1 \not\subseteq I_2$ and $I_2\not\subseteq I_1$. Then $I_1\cap I_2$ cannot be a prime ideal. Take ...
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56 views

Maximal among some ideals is prime

I am reading a lemma on noetherian integral domains but I am stuck, I am bring it up here hoping for help. The original passage is in one big fat paragraph but I broke it down here for your easy ...
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the number of non-zero integral ideals of norm m in a ring of integers [closed]

How to prove that the number of non-zero integral ideals of norm m in a ring of integers of a number field with degree n is less than or equal to the number of n-dim vectors of n positive integer ...
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46 views

On Prime and Maximal Ideals in a Commutative Ring with Unity

Let $R$ be a commutative ring with $1 \neq 0$, $I$ and $P$ are ideals of $R$. If $P$ is prime and $I \cap P \neq 0$, does it follows that either $I \subseteq P$ or $I$ is also a prime ideal ...
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How to show non unit $x$ is idempotent in $R$ if $xR+aR=R$ for all $a\in R\smallsetminus (J(R)\cup U(R)\cup\{x\})$?

Let $R$ be a commutative ring with the multiplicative identity. Let $x$ be a non unit element of $R$ such that $xR+aR=R$ for all $a\in R\smallsetminus (J(R)\cup U(R)\cup\{x\})$ where $U(R)$ is the ...
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Maximal nor prime ideal [closed]

Let $R = C([0,1])$ be the set of continuous functions from $[0,1]$ to $\mathbb R$. Consider $R$ as a ring with the following operations $(f + g)(x) := f(x) + g(x) $ and $(f.g)(x) := f(x)g(x)$ Show ...
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Quotient of noncommutative algebra

Let $R=k\langle x,y,z \rangle$ be the non-commutative algebra in $3$ variables. Let $I$ be the ideal defined by the relations $xy-yx,yz-zy,xz-zx$. How to show formally that $R/I$ is the polynomial ...
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Let $A$ be a ring. Let $I$, $J$ be two ideals of $A$.The following properties are ture. [closed]

Let $A$ be a ring. Let $I$, $J$ be two ideals of $A$.The following properties are ture. (a) The radical $\sqrt[]{\mathstrut I}$ equals the intersection of the ideals $\rho$ $\in$ V(I). (b) We have ...
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What do elements of this quotient ring look like and why?

Let $R = \{a+b \sqrt 2| a,b $ integers$\}$. Let $M = \{a+b \sqrt 2| a,b $ integers and $5|a $ and $ 5|b\}$ be its ideal. How would you write out $R/M$ in this form?
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Proving $M$ is maximal if the quotient ring $R/M$ is a field.

Let $R$ be a ring with unit element and ideal, $M$, such that $R/M$ is a field. Prove $M$ is maximal ideal. I know that because $R/M$ is a field, its only ideals are $(0)$ and itself. Also, I ...
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Show that the radical of the ideal is equal $ \langle X,Y\rangle $

$\def\Rad{\operatorname{Rad}}$ Could you give me some hints how I can solve the followig exercise? Show that the $\Rad(I)$ of the ideal $I=\langle X^5,Y^3\rangle $ of the ring $\mathbb{C}[X,Y]$ is ...
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Intersection of nonzero ideals in a right Noetherian domain is nonzero

I've been asked to show that in a right Noetherian domain, the intersection of nonzero right ideals is nonzero. A hint is given, saying that if not, then any nonzero right ideal contains a direct sum ...
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Induced Spec map for a morphism of finitely generated $\mathbb{C}$-algebras

I have a morphism $f:A\longrightarrow B$ of finitely generated $\mathbb{C}$-algebras. I have proven, using Zariski's lemma, that the inverse image of a maximal ideal $M \subset B$ is a maximal ideal ...
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Very simple question about ideals

If $A$ is a commutative unital ring, $I$ is an ideal of $A$, and $a \in A$, then I know that $$aI=(a)I$$ where $aI=\{ax : x \in I\}$ and $(a)I$ is the ideal product of the principal ideal $(a)$ and ...
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1answer
56 views

Saturation of a power of an ideal

Let $k$ be a field and let $R=k[x,y,z]$ and $\mathfrak m=(x,y,z)$. Let $I$ be a graded ideal of $R$. For all $n\in \mathbb{N}$ on has $$ (I^{\rm sat} )^n\subset (I^n)^{\rm sat},$$ where $$I^{\rm ...
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Kernel of an evaluation homomorphism $\mathbb{C}[X_1,\dots, X_n] \rightarrow \mathbb{C}$ [duplicate]

Let $R:=\mathbb{C}[X_1,\dots, X_n]$, $a=(a_1,\dots, a_n)\in \mathbb{C}^n$ and $\phi_a:R\rightarrow \mathbb{C}$, $\phi_a(f)=f(a)$. I want to show that $\ker(\phi_a)=(X_1-a_1,\dots, X_n -a_n)$. I ...
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Finding the primary decomposition of an ideal $I$ and the associated primes of $A/I$

I've the ring $R=\mathbb Z[2X,X^2,X^3$] and the ideal $I=(2X,X^2)$. I'm trying to find: the associated primes of $A/I$ and $A/I^2$, and the primary decompositions of $I$ and $I^2$. How should I ...
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The contraction of prime ideal is prime

Let $R$ be a commutative ring with identity. Let $S$ be an extension ring of $R$ and $I \neq S$ be a prime ideal of $S$. The author says that $J=I \cap R$ is a prime ideal of $R$. I fail to understand ...
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Maximal and prime ideals of quaternions with integer coefficients

Let $R = \mathbb{Z} + \mathbb{Z}i + \mathbb{Z}j + \mathbb{Z}k$, the subring of $\mathbb{H}$ consisting of quaternions with integer coefficients. An exercise in Goodearl and Warfield's An Introduction ...
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Is $I=\langle7, 3+\sqrt{19}\rangle$ a principal ideal of $\Bbb Z[\sqrt{19}]$?

Is $I=\langle7, 3+\sqrt{19}\rangle$ a principal ideal of $\Bbb Z[\sqrt{19}]$? I defined the norm : $$N(a+b\sqrt{19})=(a+b\sqrt{19})(a-b\sqrt{19})=a^2-19b^2$$ Then we can see the multiplicative ...
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Show that $\mathbb{Z}[\sqrt{223}]$ has three ideal classes.

Well the question is the title. I tried to grab at some straws and computed the Minkowski bound. I found 19,01... It gives me 8 primes to look at. I get $2R = (2, 1 + \sqrt{223})^2 = P_{2}^{2}$ $3R ...
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Proving Irreduciblity in Polynomial Quotient Rings

I'm working on an exercise from Dummit and Foote, and I've gotten down to the following lemma that makes everything I need work out, the only problem is that I'm not sure how to prove it (or whether ...
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Does every algebra automorphism preserve augmentation ideal filtration?

Let $A=\displaystyle\bigoplus_{n\geq0}A_n$ be a graded algebra and let the augmentation homomorphism $\varepsilon:A\to A_0$ be the projection. Define the augmentation ideal, denoted by $A_+$, to be ...
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Simple Maximal Ideal Question.

Question: Let $R = \{a + bi | a,b \in \Bbb Z \}$, let $M = \{x(2 + i) | x \in R\}$. Prove M is a maximal ideal of R. I just started learning about ideals so I apologize for asking a basic question, ...
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25 views

Let $I$ be a proper ideal of a ring $R$. Then $IR[\alpha_1, … , \alpha_n]$ is a proper ideal of $R[\alpha_1, … , \alpha_n]$

Let $I$ be a proper ideal of the commutative ring $R$. Then $IR[\alpha_1, ... , \alpha_n]$ is a proper ideal of $R[\alpha_1, ... , \alpha_n]$ I thought of using the fact that an ideal of any ring ...
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28 views

Is the set of pseudo-complements of the elements of an ideal in a pseudocomplemented lattice a filter?

Let $L$ be a pseudocomplemented distributive lattice with $0$ and $1$, $I \subseteq L$ an ideal and set $F = \{\neg x \; | \; x \in I\}$, where $\neg x$ is the pseudocomplement of $x$. My question is: ...
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Number Theory: Ramification

I am currently trying to figure out the following question regarding ramification. Let K = $\mathbb{Q}(\sqrt{5})$, L = $\mathbb{Q}(\sqrt{7})$, M = $\mathbb{Q}(\sqrt{35})$, and KL = ...
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Why are these two quotients equal?

I'm not being able to check why are these two quotients equal. $\mathbb C[x]/(x^2-x^3)= \mathbb C[x]/(x^2)$ Can someone tell me why is it valid?
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Ideals in a polynomial ring over a skew field

I know that a polynomial ring over a field is a PID, does this property also hold for a polynomial ring over a skew field? Is there maybe something else that characterise the ideals in that ring ?