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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How are varieties related polynomials?

My teacher says that varieties and ideals are related to each other while I tend to mix polynomials and varieties in my terminology. Could some explain how varieties are related to polynomials? And ...
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Duality between cut ideals and cycle ideals?

There exist a general duality between vertex-cuts and cycles and also Duality Principle on Digraphs. I am trying to find a duality prienciple expressed in terms of ideals so Does there exist a ...
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1answer
27 views

A homogeneous principal prime ideal in $K[x_1,\dots,x_n]$ is generated by a homogeneous element. [closed]

I expect that the following result is true, but i can't prove it. A homogeneous principal prime ideal in $K[x_1,\dots,x_n]$ is generated by a homogeneous element. I need some help to prove this....
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How to decompose that ideal?

We have $$I=\left(x^2+2y^2-3,y(x-y),y(y+1)(y-1)\right)\subset\mathbb{C}[x,y]$$ and I would like to decompose it as intersection of simpler ideals. How could I proceed? For example, in this ...
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1answer
61 views

$17\mathbb{Z}[\sqrt {10}]$ is prime ideal in $\mathbb{Z}[\sqrt {10}]$

This seems tedious since I would have to show $(a+b\sqrt {10})(c+d\sqrt {10})=17k+17j\sqrt {10}$ implies that one of the factors belongs to $17\mathbb{Z}[\sqrt {10}]$. It's easier to show something ...
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1answer
48 views

$I$ is the maximal left ideal

Let $R$ be a ring and $I\subseteq R$ the unique maximal right ideal of $R$. I have shown that $I$ is an ideal and that each element $a\in R-I$ is invertible. I want to show that $I$ is the unique ...
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1answer
38 views

Showing an isomorphism of rings

Consider the ideal $I=(1+2x)\cdot \Bbb Z[x]$ in the polynomial ring $\Bbb Z[x]$. I am trying to show that $\Bbb Z[x]/I$ is isomorphic to $R=\{\frac{a}{2^r}:a\in \Bbb Z, r\in \Bbb N_0\}$. My approach:...
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48 views

Why $\langle x^2+1\rangle$ is not prime in $\mathbb{Z}_2[x]$?

I am reading ring theory (a beginner) and I stumbled upon a problem which I can't understand The ideal $\langle x^2+1\rangle$ is not prime in $\mathbb{Z}_2[x]$, since it contains $(x+1)^2=x^2+2x+...
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1answer
27 views

$R/\langle p^k\rangle$ is an associator (i.e. if $\langle a\rangle = \langle b\rangle,$ then $a$ and $b$ are associates) when $R$ is a PID.

As the title says, I want to show that when two principal ideals are equal in $R/\langle p^k\rangle,$ where $R$ is a principal ideal domain and $p\in R$ is a prime element, then their generators are ...
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28 views

Singular ideal of an idealization

Let $S$ be a commutative ring, and let $A$ be a faithful $S$-module. Through idealization, we can make the abelian group $R=S⊕A$ into a commutative ring using the multiplication $(s,a)(s',a')=(ss',sa'+...
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29 views

Proving that there exists $a\in R$ such that $a \equiv a_k \pmod{I_k}$ [duplicate]

Let $I_1,...,I_m$ be ideals of a ring $R$ such that $I_j+\cap_{k\neq j}I_k=R$ for every $j\in\{1,...,m\}$. Then if $a_1,...,a_m\in R$ there exists $a\in R$ such that $a \equiv a_k \pmod{I_k}$ for ...
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2answers
41 views

Why is $I[x]$ not maximal $\mathbb{Z}[x]$? [duplicate]

We have that $I=(2)$ is maximal in $\mathbb{Z}$ because $(2)\subseteq (4)\subseteq \dots \subseteq (2^k)$, right? Why is $I[x]$ not maximal $\mathbb{Z}[x]$ ?
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$\mathbb{Z}[\sqrt{10}]$ is noetherian

How can we prove that $\mathbb{Z}[\sqrt{10}]$ is noetherian except by using Hilbert basis theorem? How can we find a sequence of ideals that satisfy the ACC?
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27 views

Prove that the radical of an ideal is an ideal

Let $R$ be a commutative ring with unity. For an ideal $I$ of $R$, I am attempting to prove $\sqrt{I}=\{x\,|\,x^n\in I\}$ is an ideal. Closure under multiplication with $R$ seems straight forward: ...
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30 views

Nilpotent elements lie in some prime ideal in a commutative ring with $1$

$R$ is a ring with $1$. We call $r\in R$ nilpotent if $\exists n\in \mathbb{N}$ such that $r^n=0$. Show every nilpotent element lies in some prime ideal. The fact that $1\in R$ may not be needed ...
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86 views

Ideal which becomes a principal ideal in a higher field extension

I am working on the question of why the ideal $(2,\sqrt{-6})$ is not a principal ideal in $\mathbb{Q}(\sqrt{-6})$, but becomes one in $\mathbb{Q}(\sqrt{-6},\sqrt{2})$. To prove that it is not ...
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1answer
37 views

Confusion over Grobner Bases, Division algorithm, and ideal memebership.

I'm reading through Justin Smith's Introduction to Algebraic Geometry. Before getting into coordinate rings, he talks about Grobner bases. He's given a division algorithm in which given and ordering ...
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1answer
28 views

Singular ideal containing a given nilpotent ideal

Let $R$ be a ring with identity, and $Z(R_R)$ be the singular ideal. Is it true that any nilpotent ideal of $R$ lies in $Z(R_R)$? It is well known that any central nilpotent element would belong to ...
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1answer
24 views

Restricting the quotient map of rings to a subring

When $q$ maps $R$ to $R/I$ and $p$ is the restriction of $q$ to a subring $A$ of $R$, why is the image of $p$ $(A+I)/I$? $q$ maps $r$ to $r+I$, so shouldn't $p$ map $a \in A$ to $a+I$, so image of $p$...
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2answers
53 views

Cardinal of quotient rings of gaussian integers. [duplicate]

It is known that $\mathbb{Z}[i]$ is a PID and that $\mathbb{Z}[i]/(a+bi)\mathbb{Z}[i]$ is finite for all $(a,b) \in \mathbb{Z}^2\backslash \{(0,0)\}$. My question : Is there any result on the ...
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69 views

If $I$ and $J$ are ideals in a ring $R$ with $1$ such that $I+J = R$, show that $I^m$ and $J^n$ are co-maximal for all $m,n \in \mathbb{N}$ [duplicate]

If $I$ and $J$ are ideals in a ring $R$ with 1 which are co-maximal, i.e $I+J = R$, show that $I^m$ and $J^n$ are co-maximal for all $m,n$ in $\mathbb{N}$ Work done: Should I proceed using Zorn'...
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25 views

Inverse elements in a certain monoid

Let $R$ be a ring with unity, and $Z(R_R)$ be its right singular ideal, i.e. the set of elements of $R$ whose right annihilators are essential in the right module $R_R$. My question: If $x\in Z(...
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48 views

Intersection and product of principal ideals

I'm trying to show that $(x)\cap(p^n)=(xp^n)$, where $x,p$ are elements of some ring $R$, $p$ is prime and $p\nmid x$. The inclusion from right to left is obvious, but I can't make any progress in ...
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1answer
86 views

How to compute $\dim_{\mathbb C}\mathbb{C}[x,y,z]/(z^4,x^2+y^2+z^2-1,xy)$?

How to compute $\dim_{\mathbb C}\mathbb{C}[x,y,z]/(z^4,x^2+y^2+z^2-1,xy)$? I tried to decompose $$(z^4,x^2+y^2+z^2-1,xy)=(z^4,x^2+y^2+z^2-1,x)\cap(z^4,x^2+y^2+z^2-1,y)=(z^4,x^2+z^2-1,y)\cap(z^4,y^2+...
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62 views

Let $R$ be a ring, $S$ a subring and $I$ an ideal. If $R$ is Noetherian, are then $S$ and $R/I$ also Noetherian?

Let $R$ be a ring, $S$ a subring and $I$ an ideal. If $R$ is Noetherian are then $S$ and $R/I$ also Noetherian? I have done the following: $R$ is Noetherian iff each increasing sequence of ideal $...
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1answer
42 views

Stabilization of colon ideals of a decomposable ideal

Let $\mathfrak a$ be a decomposable ideal in a (commutative ring with unity) $A$, let $\Sigma$ be an isolated set of prime ideals belonging to $\mathfrak a$, and let $\mathfrak q_\Sigma$ be the ...
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1answer
84 views

Maximal ideals of the ring $R=C(\mathbb R)$ of continuous functions

Let $R=C(\mathbb R)$ be the ring of continuous functions $f:\mathbb R\to\mathbb C$ where the addition and the product is pointwise defined. Let $$\mathbb m_a=\{f\in R\ |\ f(a)=0\}$$ be a maximal ...
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1answer
88 views

Why is $(x,y)\cap(x,z)\cap(x,y,z)^2$ a minimal primary decomposition of $(x,y)(x,z)$?

Why is $(x,y)\cap(x,z)\cap(x,y,z)^2$ a minimal primary decomposition of $(x,y)(x,z)$? I understand that the ideals are primary and also that one has $$(x,y)\cap(x,z)\cap(x,y,z)^2=(x,y)(x,z).$$ But I ...
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1answer
42 views

Support of quotient sheaf of ideal sheaves with same support

I'm not very sure about this argument. Let $\mathscr{I},\mathscr{J}$ two ideal sheaves (you can think about ideal sheaves over a projective variety or even the projective space itself) and assume that ...
3
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1answer
32 views

Examples of non-ideals

For a subgroup $(I,+)$ of additive group part of ring $(R,+,•)$ to be an Ideal we need $a•I$ and $I•a$ to be subset of $I$ for all $a$ in $R$. The exercises in my textbook gives a lot of examples of ...
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We Quotient an algebraic structure to generate equivalence classes?

Till now I visualized Quotient groups as a technique to generate equivalence classes whenever needed, as in $\mathbb{Z}/n\mathbb{Z}$. But now I have a feeling of doubt since I haven't seen a book that ...
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31 views

Basic ideals exercise, and a question about notation definition

From this this book. Given a finite set $\left\{f_1,f_2,\ldots,f_r\right\} \subset R$, the ideal $I$ generated by this set is denoted $f_1, f_2, \ldots , f_r$ and consists of all the sums $f_1h_1 ...
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1answer
28 views

Showing $r(\mathfrak a)=(1)$ iff $\mathfrak a=(1)$

I am stuck on the following exercise: Suppose $\mathfrak a$ is an ideal of a ring $A$ (commutative with unity) and $r(\mathfrak a)=\{x\in A\mid\exists n \in \mathbb{N}: x^n \in \mathfrak a\}$. I ...
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4answers
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Show that $\langle3\rangle$ is a maximal ideal in $\mathbb{Z}[i]$ [closed]

Equivalently how can I show that $\mathbb{Z}[i]/\langle 3\rangle$ is a field?
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1answer
55 views

Prove that P is prime ideal of R?

$R$ is a commutative ring and $1\in R$. Let $I$ be an ideal of $R$ and $P$ be a prime ideal of $I$. Then show that $P$ is prime ideal of $R$. I know how to prove that P is ideal of R. Suppose that $...
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1answer
48 views

Is it correct this way to compute that radical ideal?

Is it correct to compute that radical ideal in this way? $$\sqrt{(x^2,xz^2-x,y-z)}=\sqrt{(x^2,xz^2-x,y-z,x)}=\sqrt{(y-z,x)}=(x,y-z)$$ In particular, I added $x$ to generators inside the 'root' ...
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1answer
56 views

Radical of an ideal.

How can I compute $\sqrt{(x^2+y^2-1,yz-1)}$ as ideal of $\mathbb{C}[x,y,z]$? Actually I have to prove that $(x^2+y^2-1,yz-1)$ is prime but I don't know how. Could you give me some suggestions, ...
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1answer
25 views

Understanding the definition of $UV$, where $U$ and $V$ are ideals in a ring

I have the following question at hand: I. N. Herstein Topics in Algebra: Ideals and Quotient Rings : Qn $3.4.6$ If$\ \ U,V$ are ideals of $\ R\ $,let $UV$ be the set of all elements that can be ...
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Left ideals of $M_2(K)$ with $K$ a field

Is it true that the only proper left ideals of $M_2(K)$, the ring of the matrices whose coefficients are in a field $K$, are $$ \left\{\begin{pmatrix}ah & ak \\ bh & bk\end{pmatrix}: a,b \in K\...
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1answer
46 views

Is it true that $I+J\subseteq I\cap J$ [closed]

Let $R$ be a commutative ring and $I,J$ are ideals in $R$. Is it true that $$I+J\subseteq I\cap J?$$ I know that $I+J =\{x+y|x\in I, y\in J\}$.
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1answer
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Number of distinct prime ideals in $\mathbb {Q}[x]/\langle x^m-1 \rangle$ [closed]

Number of prime ideals in quotient ring obtained by $\mathbb {Q}[x]/\langle x^m-1 \rangle$ is ...?
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1answer
28 views

Essential Prime Ideal

I search for an example of a commutative ring $R$ with unity having a prime ideal $P$ and some element $r\in R$ such that the annihilator of $r$ is both contained in $P$ and essential in $R$. By ...
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2answers
34 views

Why Is $I$ A Maximal Ideal If $R/I$ Is A Field?

How can I prove that if $R/I$ is a field then $I$ is a maximal ideal? In my book it says that this is a corollary of the following theorem: If $φ:\mathbb{F}→S$ is a non trivial homomorphism and $\...
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1answer
18 views

Some equivalences for Ideals of the ring of real valued functions

Let $I_{N}=\{f\in\Omega(\Bbb R,\Bbb R)|\forall x \in N\subset \Bbb R: f(x)=0\}$, where $\Omega(\Bbb R,\Bbb R)=\{f:\Bbb R \to \Bbb R|f\space is \space a\space function\}$, then the following statements ...
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1answer
27 views

Why the ideal norm is multiplicative

Let $I\subseteq B$ be an ideal, we define the ideal norm of $I$ as the ideal in $A$ generated by the elements $N_{E/K}(\alpha)$ where $\alpha \in I.$ We denote it by $N_{E/K}(I).$ If $\mathfrak{p}$ ...
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35 views

Question about radical ideal

Suppose $\mathbb{R}[X]$ is the normal multivariate polynomial ring where $X = x_1,...x_n$. $\mathbb{R}[X]_t$ is the truncated set such that $\mathbb{R}[X]_t =\left\{f: f \in \mathbb{R}[X], \deg(f) \...
2
votes
2answers
52 views

Given two polynomials $f$ and $g \in \mathbb{Q}[X]$, prove that $(f) + (g) = (h)$ and $(f)\bigcap(g) = (k)$

Given two polynomials $f(X) = 3X^2 + 7X - 6$ and $g(X) = 2X^2 + 5X - 3 \in \mathbb{Q}[X]$, prove that there exist $(h)$ and $(k) \in \mathbb{Q}[X]$ such that $(f) + (g) = (h)$ and $(f)\bigcap(g) = (k)$...
1
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1answer
44 views

Saturation of homogeneous ideal

Let $I \subset S=k[x_0,...,x_n]$ be a homogenous ideal. The saturation of $I$, $\bar{I}$ is defined to be $\{s \in S: \exists m \; s.t. \; \forall i \; x_i^m s \in I\}$ Is it true that $\bar{I}=(s \...
0
votes
2answers
31 views

Showing Quotient ring is a field using maximal Ideal

Question: Show that $R\left [ x \right ]/\left \langle x^{2}+1 \right \rangle$ is a field. Recall: Theorem: Let R be a commutative ring R with unity. Let I be a proper Ideal of a ring R. ...
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1answer
8 views

Verify size of factor ring

Let the ring $R=\left \{ \begin{bmatrix} a_{1} &a_{2} \\ a_{3}& a_{4} \end{bmatrix} \mid a_{i} \in \mathbb{Z} \right \}$ and let I be the subset of R consisting of matrices with even ...