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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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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23 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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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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$I$ 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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36 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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41 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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43 views

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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36 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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61 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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59 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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53 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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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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50 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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25 views

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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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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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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70 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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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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35 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
220 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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1answer
45 views

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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1answer
52 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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1answer
12 views

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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1answer
40 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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1answer
49 views

How to show non unit $x$ is idempotent in $R$ if $xR+aR=R$ for all $a\in R\smallsetminus (J(R)\cup U(R))$?

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 non unit $a$ outside the Jacobson radical of $R$, i.e., For all $a\in ...
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1answer
71 views

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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1answer
47 views

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 ...
2
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1answer
39 views

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 ...