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 do I find the ideal $I+J$?

This is a homework problem: Consider the polynomial ring $R=\mathbb Z_2[x_0,x_1,\dots,x_n]$. Let $I=\langle x_0x_1\cdots x_n\rangle$ and $J=\langle x_0+x_1,x_0+x_2,\dots,x_0+x_n\rangle$. Find $I+J$...
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0answers
7 views

Relationship between ST-Path Ideals and ST-Cut Ideals?

Topic: st-connectivity, st-cut ideals and path ideals of a graph My Lemma: None of the st-cut-monomials vanish iff there is at least a st-path that does not vanish. Example ST-cuts: {{1,3,5,6},{...
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1answer
21 views

Showing that $\mathrm{Rad}((0)) ≠ (0)$ implies $R^\times \subsetneq R[X]^\times$

Let $R$ be a commutative ring with $1$, and let $I ≤ R$ be an ideal. We call $\mathrm{Rad}(I) := \{r \in R: \exists n \in \mathbb{N}_0: r^n \in I\}$ the radical of $I$. I now want to show that if $\...
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1answer
84 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 ...
0
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2answers
71 views

How to decompose that ideal? [on hold]

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 ...
3
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1answer
397 views

if $S$ is a ring (possibly without identity) with no proper left ideals, then either $S^2=0$ or $S$ is a division ring.

I'm having trouble with this homework problem (from Algebra by Hungerford). If $S$ is a ring (possibly without identity) with no proper left ideals, then either $S^2=0$ or $S$ is a division ring. ...
3
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1answer
100 views

In a commutative ring without identity, is $(a)(b)\subset (ab)$ or $(ab)\subset (a)(b)$?

Let $R$ be a commutative ring without unity. Consider an ideal $(a)$ generated by $a\in R$. Note that $(a)=\{ra+na : r\in R, n\in \textbf Z\}$ since $R$ has no identity. I wonder if $(a)(b)\subset (ab)...
2
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1answer
46 views

A prime ideal which is not maximal

I am searching for a prime ideal of the ring $R=∏_{n=2}^{∞} {\mathbb Z}_{2^n}$ which is not maximal. In fact, since each ${\mathbb Z}_{2^n}$ is local with $\left<\bar 2\right>$ as the maximal ...
3
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0answers
41 views

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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16 views

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 ...
0
votes
1answer
84 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+...
1
vote
5answers
309 views

Maximal ideal in the ring of polynomials over $\mathbb Z$

Let $\mathbb Z[x]$ the ring of polynomials with integers coefficients in one variable and $I =\langle 5,x^2 + 2\rangle$, how can I prove that $I$ is maximal ideal. I tried first see that $5$ and $x^...
19
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5answers
7k views

Why are maximal ideals prime?

Could anyone explain to me why maximal ideals are prime? I'm approaching it like this, let $R$ be a commutative ring with $1$ and $A$ be a maximal ideal. Let $a,b\in R:ab\in A$ I'm trying to ...
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1answer
24 views

Why does this hold in these cases? [on hold]

Let $R$ be a U.F.D. and $0\neq d\in R$. If $d\notin U(R)$ do we have that $d=a_1^{k_1}\dots a_r^{k_r}$ with $a_i$ irreducible? If $d\in U(R)$ why does it hold $(d)=R=(1)$ ?
0
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1answer
56 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 ...
0
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1answer
47 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 ...
0
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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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2answers
47 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
24 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 ...
3
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1answer
84 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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0answers
26 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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5answers
1k views

If $I = \langle 2\rangle$, why is $I[x]$ not a maximal ideal of $\mathbb Z[x]$, even though $I$ is a maximal ideal of $\mathbb Z$?

Let $I = \langle 2\rangle$. Prove $I[x]$ is not a maximal ideal of $\mathbb Z[x]$ even though $I$ is a maximal ideal of $\mathbb Z$. My professor mentioned that I should try adding something to it ...
0
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0answers
27 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 ...
3
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1answer
672 views

How do we find the prime ideals of a ring of integers of a number field?

For example, for $F=\mathbb Q(\sqrt{-5})$ the ring of integers of $F$ is $\mathbb Z[\sqrt{-5}]$ (since $-5\equiv3 \pmod 4$). How can we determine the prime ideals of this ring? Another problem is the ...
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2answers
38 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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0answers
40 views

$\mathbb{Z}[\sqrt{10}]$ 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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0answers
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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2answers
29 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 ...
2
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2answers
53 views

To show that $\langle x-a , y-b\rangle$ is a maximal ideal of $F[x,y]$ by showing that $F[x,y]/\langle x-a , y-b\rangle$ is a field [duplicate]

Is there any way to show that for $a,b \in F$ , the ideal $\langle x-a , y-b\rangle$ is maximal in $ F[x,y]$ , by showing that the quotient $F[x,y]/\langle x-a , y-b\rangle$ is a field ? Is the ...
1
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1answer
103 views

Does validity of Bezout identity in integral domain implies the domain is PID? [closed]

Let $D$ be an integral domain such that for any $a,b \in D$, $Da+Db$ is a principal ideal. Then must $D$ necessarily be a principal ideal domain i.e. should all the ideals of $D$ be principal ?
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2answers
32 views

Showing the kernel of $f:\mathbb{Z} \rightarrow \mathbb{Z}_n, f(x)=[x]_n$ is the ideal $n\mathbb{Z}$? [closed]

How to display that the kernel of $f:\mathbb{Z} \rightarrow \mathbb{Z}_n, f(x)=[x]_n$ is the ideal $n\mathbb{Z}$? I cannot understand from reading "the kernel is $n\mathbb{Z}$" that it really is the ...
2
votes
1answer
26 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 ...
21
votes
5answers
1k views

Why are ideals more important than subrings?

I have read that subgroups, subrings, submodules, etc. are substructures. But if you look at the definition of the Noetherian rings and Noetherian modules, Noetherian rings are defined with ideals ...
0
votes
1answer
35 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 ...
0
votes
1answer
86 views

A problem on maximal ideal in polynomial ring.

Let $\Bbb R[x]$ be the polynomial ring over $\Bbb R$ in one variable. Let $I\subseteq\Bbb R[x]$ be an ideal. Then which are true? $I$ is a maximal ideal if and only if $I$ is a non-zero prime ...
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0answers
31 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) \...
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1answer
22 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 $p$...
0
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2answers
46 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 ...
1
vote
1answer
88 views

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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votes
2answers
65 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'...
0
votes
0answers
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(...
1
vote
1answer
39 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 ...
0
votes
0answers
46 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 ...
0
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0answers
68 views

The ring $R=C(\mathbb R)$ of continuous functions $f:\mathbb R\to\mathbb C$

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 ...
0
votes
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' ...
1
vote
2answers
53 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 $...
2
votes
1answer
40 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
votes
1answer
31 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 ...
3
votes
2answers
55 views

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 ...
0
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0answers
42 views

Normal Subgroups and Ideals

For a subgroup $(N,+)$ of group $(G,+)$ to be Normal we need $a+N=N+a$ for all $a$ in $G$. For a subgroup $(I,+)$ of additive group part of ring $(R,+,•)$ to be Ideal we need $a•I$ and $I•a$ to be ...