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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Build a reduced ring starting from an ordinary one

This may be easier than I think, but still I can't seem to wrap my head around it. I've learnt that if we take a ring $A$ and quotient it for a (two-sided) ideal $I \subset A$ which is radical, the ...
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41 views

Ideals of Unique Factorization Domain

Let R be a commutative ring with unity such that R[x] is UFD. The ideal (x) of R[x] is denoted by I. Then pick the correct statements from below: 1. I is prime. 2. If I is maximal then R[x] is a PID. ...
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Let $R$ be a commutative ring. For $a \in R$ consider the set $(a) = \{r*a | r\in R\}$. Show that $(a) = R$ if a is a unit of $R$ [duplicate]

I tried some values and I think I got the idea. R is the set of values used in the ring. If I use $\mathbb{Z}$, the units are $\{-1,1\}$. If I take 1 for example, I could use it to get every value in $...
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Decomposition of a monomial ideal

I have to find a primary decomposition of the following ideal and I proceeded in this way: $$(x^2z,x^2y^3,xt^2)=(x)\cap(t^2,x^2z,x^2y^3)=(x)\cap(t^2,x^2)\cap(t^2,z,z^2y^3)=(x)\cap(t^2,x^2)\cap(t^2,z,...
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When a prime ideal in polynomial ring over integers is principal [duplicate]

While dealing with a question about a prime ideal $I\subset\mathbb{Z}[x]$ (with $0$ in $I$ as the only constant polynomial) I was asked to show that there exist $f(x)\in\mathbb{Z}[x]$ such that $I=\...
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45 views

Finding nilpotent elements in a quotient ring.

Which are nilpotent elements of $\mathbb{Q}[x]/(x^5-3x^2)\times\mathbb{Z}/(12)$? I tried to decompose in this way: $$\mathbb{Q}[x]/(x^5-3x^2)\times\mathbb{Z}/(12)\cong\mathbb{Q}[x]/(x^2)\times\mathbb{...
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Principal ideal of $\mathbb{C}[Z,\bar{Z}]$

Let $I$ be an ideal of $\mathbb{C}[Z,\bar{Z}]$. How to prove that $I$ is principal in $\mathbb{C}[Z,\bar{Z}]$ ? It exists some simple criterion to say that an ideal will be principal or not?
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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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How do I find the ideal $I+J$ and quotient $R/(I+J)$?

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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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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85 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 ...
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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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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 ...
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75 views

How to decompose that ideal? [closed]

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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Why does this hold in these cases? [closed]

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 $(d)=R=(1)$ ?
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$17\mathbb{Z}[\sqrt {10}]$ 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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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 ...
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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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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 $(x+1)^2=x^2+2x+...
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$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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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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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 ...
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40 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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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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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 ...
3
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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
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 ...
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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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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 $p$...
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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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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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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 ...
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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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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
41 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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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 ...
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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
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 ...
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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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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 ...
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2answers
56 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 ...
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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
59 views

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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Is this localization of ideal correct?

If I have $$I=\left(x^2+y^2-yz,xyz-x,y(y-z)(yz-1)\right)\subset\mathbb{C}[x,y,z]$$ is then $I\mathbb{C}[x,y,z]_{(x,y)}$ equal to $\left(x^2+y^2-y,x,y\right)=\left(x,y\right)$? Thank you!
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1answer
53 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 $...
0
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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' ...