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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If a certain ideal is radical or not

Let $n \in \mathbb{N}$ and let $I_{n}$ be an ideal in the polynomial ring $\mathbb{C}[x_{1},...,x_{n}]$ with the following properties: I is generated by a (finite) number of polynomials which are ...
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Minimal right ideals

Let $I$ be a minimal right ideal of a ring $R$ with $1$. If $r\in R$, could we say that $rI$ is zero or a minimal right ideal? I assumed a right ideal $J$ in $rI$ and intersecting it with $I$ got a ...
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If $n|m$ prove that the natural surjective projection $\pi: \mathbb{Z_m} \rightarrow \mathbb{Z_n}$ is also surjective in units

Not sure if this is the right proof: since $n|m \Rightarrow n \leq m$, then if we factor $m = p_1^{\alpha_1}p_2^{\alpha_2}\ldots p_k^{\alpha_k}$, then $n = p_1^{\beta_1}p_2^{\beta_2}\ldots ...
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Definition of $\sum_{a \in A} I_a$

If $(I_a)_{a \in A}$ a family of ideal of $K[x_1,x_2, \dots, x_n]$, I have the following definition in my notes: $$\sum_{a \in A} I_a=\{ a_{i1}+a_{i2}+ \dots+ a_{ij} | a_{ij} \in I_{a_j} \}$$ Is ...
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Fractional ideals of $\mathbb{Q}$ prime to $N$

Let $N \in \mathbb{Z}$. What is meant by a fractional ideal $\mathfrak{p}$ of $\mathbb{Q}$ being prime to $N$? Is it that $gcd(\mathfrak{p},N\mathbb{Z})$ contains $\mathbb{Z}$? Let $I_N$ denote the ...
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algebra with topology homework problem

Hello Everyone, I have this homework problem, I'm going to share what i have so far, not sure if Im in the right path. First, I have: $$f \sim g \, \Leftrightarrow \,x_0 \in \mathbb{R^n}, \exists ...
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14 views

Carlson's translatability - are theses characterisations equivalent?

Given a translation-invariant ideal $\mathcal{I}$ on a commutative group $G$ and it's dual filter $\mathcal{I}^*$, I am trying to show that $$ (\forall I \in \mathcal{I})(\exists I' \in ...
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14 views

$M_n(D)$ is left and right-simple?

Is it true that if $D$ is a division ring and $n\in\mathbb{Z}_{\geq1}$, then the only left and right ideals of the ring $M_n(D)$ are the trivial ones? I know that $M_n(D)$ is simple, and the ...
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24 views

Correspondence between ideals of $R$ and $D^{-1}R$

Let $R$ be an integral domain, and $D\subset R$ be a multiplicatively closed subset such that $1\in D$ and $0\not\in D$ . Prove/disprove that there is a one-to-one correspondence between the ideals of ...
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How to show that an ideal of $F[x]$ containing an irreducible polynomial of degree $n$ and a nonzero polynomial of degree $<n$ is $F[x]$?

Let $F$ be a field and suppose that $I$ is an ideal of $F[x]$ which contains an irreducible polynomial of degree $n$ and a nonzero polynomial of degree less than $n$. Show that $I=F[x]$. I can't ...
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Showing that an ideal is principal.

I need to show that the ideal $(3 + i , 6)$ is principal in $\mathbb{Z}[i]$ and find its generator. So I know that I need to find an element such that $<t> = (3 + i , 6)$. My intuition tells me ...
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27 views

Questions about ring of smooth functions

First of all, this is a homework problem. Let $C^{\infty}(\mathbb{R})$ denote the ring of smooth functions. Let $I_n$ denote the set of $f\in C^{\infty}(\mathbb{R})$ such that $$f^{(k)}(0)=0, \ 0 ...
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31 views

Matrix rings and ideals

How would one go about checking if a given 2 x 2 matrix is an ideal. I am unclear as to what an ideal is and would like to know the steps in order to make the verification. Also, if it helps, I had ...
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20 views

Comparing an ideal and its saturation

Let $S = k[x_0,x_1,\ldots,x_n]$ with its usual grading and let $I \subset S$ be a homogeneous ideal not containing $S_+ = (x_0,x_1,\ldots,x_n)$. We define the saturation of $I$ to be the homogeneous ...
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19 views

$I$ is an ideal in $R$ implies that $I[x]$ is an ideal in $R[x]$.

Is the following statement right? If $I$ is an ideal in the ring $R$, then $I[x]$ is an ideal in the polynomial ring $R[x]$. If so, how can I prove it?
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Quotient ring of the ring of integers of an algebraic number field and its fraction field

Let $K$ be an arbitrary algebraic number field. We know that the fraction field of $\mathcal{O}_K$ is $K$ which is always isomorphic to some $\mathbb{Z}[x]/(f(x))$. $\mathcal{O}_K$ also has dimension ...
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Every ideal is contained in a prime ideal that is disjoint from a given multiplicative set

Let $R$ be a ring $I\subset R$ an ideal and $S\subset R$ be a set for which holds: $1)$ $1\in S$ 2) $a,b \in S\Rightarrow a\cdot b\in S$ Show that there exists a prime ideal $P$ in $R$ containing ...
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41 views

About images of (prime) ideals under injective endomorphisms

Let $f : R \to R$ be an injective unitary endomorphism of a commutative ring with 1. Let $I$ be an ideal of $R$. I have several related questions concerning the image of $I$ under $f$: 1) Under which ...
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25 views

How to show that if $\gamma \alpha=0$ (mod $A$) then $\alpha = 0$ (mod $A$)

Let $A$ be an ideal of $\mathcal O$ (Ring of integers of some algeibraic number field) and assume that $gcd([\gamma],A)= [1]$. How to show that if $\gamma \alpha=0$ (mod $A$) then $\alpha = 0$ (mod ...
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45 views

Prove $\text{rad}(I)/I$ is isomorphic to $\mathfrak{N}(R/I)$

I want to know if this is the correct way to do it. Define $\varphi:\text{rad}(I) \longrightarrow \mathfrak{N}(R/I)$ by $\varphi(r)= r^n+I$,then ker$\varphi = I$, so therefore by the 1st isomorphism ...
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Prove every prime ideal of a ring is a radical ideal.

this is my attempt: Since $R$ is commutative, we let $I$ to be a prime ideal of $R$, the for $a,b\in R$,then the product $ab$ we must have that $a\in I$ or $b \in I$, by definition of a prime ideal. ...
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45 views

Prime ideals of the ring of integers of an algebraic number field

I am working on a problem that has a completely different point and I didn't work with algebraic number fields much before, so I was wondering if someone could point me in the right direction for ...
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28 views

Help unmasking a disguised principal ideal

I recently saw a question on here about trying to generate a non-principal ideal in a principal ideal domain, with the only answer so far saying that if the ring $R$ is a PID, then $\langle e, f ...
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70 views

Show that $V(I \cap J)=V(I) \cup V(J)$.

Let $I$, $J$ ideals of $K[x_1, x_2, \dots , x_n]$. I want to show that $$V(I \cap J)=V(I) \cup V(J)$$ I tried the following: $$\subseteq: $$ Let $x \in V(I \cap J)$. From the definition of $V$: ...
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60 views

Set of roots of sum is equal to the intersection [closed]

Let $(I_a)_{a \in A}$ be a family of ideals of $K[x_1,x_2, \dots, x_n]$. I want to prove that: $$V \left ( \sum_{a \in A} I_a\right )=\bigcap_{a \in A} V(I_a)$$ Do we have to use the definition: ...
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55 views

Show that there are two ideal classes in $\mathbb{Z}[\sqrt{10}]$

Show that there are two ideal classes in $\mathbb{Z}[\sqrt{10}]$. I'm trying this problem with the Minkowski bound, please I need more help. Thanks
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30 views

Finding all ideals in $\mathbb{Q}[x]/I$, where $I$ is the ideal generated by $p(x)=x^2(x^2+x+1)$

I want to find all ideals in $\mathbb{Q}[x]/I$, where $I$ is the ideal generated by $p(x)=x^2(x^2+x+1)$. I know that $$\mathbb{Q}[x]/I \cong \mathbb{Q}[x]/(x^2) \times \mathbb{Q}[x]/(x^2+x+1)$$ I ...
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In a PID, does every attempt to generate a non-principal ideal just lead back to the whole ring itself?

It is a well-known fact that a unique factorization domain is a principal ideal domain, in which all ideals are principal ideals. [EDIT: I got dyslexic on this one, should've said something along the ...
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51 views

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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29 views

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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Quotient by the ideal $I=(2\sqrt{2})$ in the ring $R=\mathbb Z[\sqrt{2}]$

Let $R=\mathbb Z[\sqrt{2}]$ and $I=(2\sqrt{2})$ be the principal ideal generated by $2\sqrt{2}$. Let $a,c\in \{0,1,2,3\}$ and $b,d \in \{0,1\}$ and suppose that $$a+b\sqrt{2}+1=c+d\sqrt{2}+I$$ ...
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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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175 views

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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1answer
71 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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44 views

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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62 views

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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1answer
36 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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2answers
25 views

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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57 views

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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1answer
49 views

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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1answer
222 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 ...