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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Prove that the sum of ideals of a ring A equals A and its intersection is zero.

I've been looking at a couple of ring theory exercises and there's this one I don't know how to do it. It goes like this. $A$ is a commutative unital ring, and $e$ an element of $A$, $e \neq ...
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1answer
36 views

Saturation of a multiplicatively closed subset

Exercise 3.7 of Atiyah-MacDonald asks the reader: if $A$ is a commutative ring and $\mathfrak{a} \triangleleft A$ an ideal, find the saturation of $1 + \mathfrak{a}$. Previously we have shown that ...
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1answer
32 views

Property of multiplication of ideals in $\mathcal{O}_K$

Let $\mathfrak{a}, \mathfrak{b}$ be two coprime ideals of $\mathcal{O}_K = \mathbb{Z}[\sqrt{-d}]$ such that $\mathfrak{a}\mathfrak{b} = (n)$ for some $n \in \mathbb{Z}$. Does $\mathfrak{a}^m = (u)$ ...
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45 views

Generators of the Tangent Space

Let $X$ be an affine variety, $X \subset A^n$ and suppose $f_1(T),\ldots,f_r(T) \in K[T_1,\ldots,T_n] $ generate $I(X)$. (Note that $I(X)$ is the ideal of $K[T_1,\ldots,T_n]$ of which elements of $X$ ...
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47 views

Is there any way to gain some insight into a proof by simply looking at a graphic?

My school is using Pinter's "A Book of Abstract Algebra" for both semesters of Modern Algebra. For a class assignment a couple weeks ago, regarding rings, I was tasked with the following problem ...
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If $X$ is maximal ideal then it consists of non-invertible elements?

I'm reading through a paper where I came across the following theorem Let $A$ be a commutative complex Banach algebra with unit element $e$. Theorem: A subspace $X\subset A$ of codimension $1$ is ...
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44 views

Essential ideals

I am trying to get my head around essential ideals. In literature I found 2 definitions: An ideal $I$ in a C*-algebra $A$ is essential in $A$ (i) if $aI = 0$ implies $a=0$, $a\in A$; or (ii) if ...
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25 views

Local banach algebra without zero divisors

I need to construct example of banach algebra with unique nontrivial maximal ideal and without zero divisors. I think it is must be a subalgebra of $\mathbb{C}[[z]]$, but I could not build anything.
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1answer
95 views

Computing the radical of $\mathfrak{gl}(2,\mathbb{C})$ without using the semisimplicity of $\mathfrak{sl}(2,\mathbb{C})$.

I have been trying to show that the radical of $\mathfrak{gl}(2,\mathbb{C})$ is its center, i.e. scalar matrices, however all the proofs I have encountered (e.g. Radical of $\mathfrak{gl}_n$) have ...
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71 views

Extended ideals and algebraic sets

Let $L\subset k$ a field extension such that $k$ is algebraically closed. Now consider the algebraic set $Z(\mathfrak a)$ where $\mathfrak a$ is an ideal of $k[T_1,\ldots, T_n]$ but it is generated ...
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34 views

Maximal chain of primes in a finitely generated $\mathbb C$-algebra

Let $A=\mathbb{C}[x,y,z]/\langle xyz-1\rangle$. Find a maximal chain of primes in $A$. I think it has to do something with the Krull dimension but I don't really know how to construct such a ...
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1answer
56 views

Given $I,J$ ideals of $R$, show that $\forall m \geq 1; I^m + J^m =R$ [duplicate]

Given $R$ a commutative ring and $I,J$ ideals of $R$, such that $I+J=R$, show that $I^m+J^m=R, \forall m \geq 1$ My problem is that I don't know what is the meaning of $I^m$ and in literature I ...
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1answer
30 views

Ideal intersection in Boolean polynomial ring

I'm trying to calculate $ \langle x,y,z\rangle \cap \langle x+1,y+1,z+1\rangle $ in the ring of Boolean polynomials. In CoCoA initially I set the ring as ...
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37 views

Prove there isn't an isomorphism between quotient polynomial rings

Prove there isn't an isomorphism $$\phi: {{\mathbb Q [x]} \over {I_1}} \to {{\mathbb Q [x]} \over {I_2}}$$ when $I_1=\langle x^2-2\rangle$, $I_2=\langle x^2+2\rangle$. I want to assume there is an ...
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1answer
15 views

$IM$ not finitely generated , $J \subseteq I$, $JM$ finitely generated; is there some $a\in I$ such that $JM\subsetneq\langle a,J\rangle M$?

Let $R$ be a commutative ring with unity, $M$ be an $R$-module, $I$ be an ideal of $R$ such that $IM$ is not a finitely generated submodule. Let $J \subseteq I$ be a finitely generated ideal such that ...
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1answer
29 views

$M$ be a finitely generated $R$-module , and $N$ be a submodule of $M$ ; is it possible to have a meaning for $Ann(M)/N$ as an ideal?

Let $M$ be a finitely generated $R$-module, and $N$ be a submodule of $M$; is it possible to have a meaning for $Ann(M)/N$ as an ideal? (I ask this question due to its use in the third line in the ...
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1answer
40 views

Find a nontrivial proper ideal of $\mathbb{Z}\times\mathbb{Z}$ that is not prime

So I know that $4\mathbb{Z}\times\mathbb{Z}$ is a non-prime ideal of $\mathbb{Z}\times\mathbb{Z}$, and why it is. My question is, how would you find this without testing out many different ideals of ...
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30 views

An Ideal in the Group Ring $RG$

So I'm working on Abstract Algrebra (Dummit & Foote). Let $R$ be a commutative ring with identity $1$ and let $G=\{g_1, ..., g_n\}$ be a finite group. Prove that $$I=\{\sum_{i=1}^n ag_i | a\in ...
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69 views

Find all prime ideals and maximal ideals of $\mathbb{Z}/12\mathbb{Z}$

How do I go about finding these? I know that the prime and maximal ideals in this case are the same, and that an ideal $M$ is only a maximal ideal of $R$ iff $R/M$ is a field, but I don't really know ...
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1answer
31 views

Finding a prime in a ring extension using Nakayama's lemma

This is a follow up to my previous question here if $A \subset B$ is a finite ring extension and $P$ is a prime ideal of $A$ show there is a prime ideal $Q$ of $B$ with $Q \cap A = P$. (M. Reid, ...
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Given ring $F[X]/(X^2)$ why is the ideal (X) the unique maximal ideal of the ring [duplicate]

Given ring $F[X]/(X^2)$ I'm trying to understand why the ideal (X) is the unique maximal ideal of the ring. I have figured out that an element in the ring is either in the ideal (X) or is a unit, but ...
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Showing there is a prime in a ring extension using Nakayama's lemma

Here's the problem that I'm working on: if $A \subset B$ is a finite ring extension and $P$ is a prime ideal of $A$ show there is a prime ideal $Q$ of $B$ with $Q \cap A = P$. (M. Reid, ...
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1answer
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Subring of a field [closed]

Let $R$ be a subring of a field $F$ such that for each $x\in F$ either $x\in R$ or $x^{-1}\in R$. Prove that if $I$ and $J$ are two ideals of $R$, then either $I\subseteq J$ or $J\subseteq I$.
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nil Jacobson radicals

Let $f$ be an idempotent element in a ring $S$ with Jacobson radical $J$ so that both $fJf$ and $(1-f)J(1-f)$ are nil. I guess that $J$ is nil too, but I am not sure. I know that the former is ...
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1answer
43 views

Need help giving Abstract Algebra Examples relating to Ideals and Quotient Groups.

I'm having a hard time trying to provide an example of the following two problems: An example of a ring $R$ and an ideal $I$ of $R$ such that neither $R$ nor $I$ has zero divisors but the quotient ...
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1answer
109 views

Ideal of $\mathbb{C}[x,y]$ not generated by two elements

Consider the ring of polynomials in two variables $\mathbb{C}[x,y]$. Show that the ideal $\langle xy^3, x^2y^2, x^3y\rangle$ cannot be generated by two elements. Until now, I assumed by ...
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Is multiplicative inverse defined for ideal? Eg. $x^3 y\in \langle x^3 y^2\rangle$?

Definition. A subset $I\subset k[x_1,\ldots,x_n]$ is an ideal if i. $0\in I.$ ii. If $f,g\in I$, then $f+g\in I$. iii. If $f\in I$ and $h\in k[x_1,\ldots,x_n]$, then $hf\in I$. ...
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Ideal $K$ in ring of germs is generated by nonnegative functions

I asked a question about details allowing to answer this question earlier today. Unfortunately, I didn't manage to complete the exercise. Since the other questions were about another problem, I write ...
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Ring of germs of smooth functions on $\mathbb{R}^{n}$ in $0$.

First of all, I'm quite new to this theory, so it may be very dumb questions. Sorry for that. Let $R$ be the ring of germs of $C^{\infty}$ functions on $\mathbb{R}^{n}$ in $0$. Let $K$ be the ...
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Is $\mathbb{Z}[\sqrt{15}]$ a UFD?

Let $R=\mathbb{Z}[\sqrt{15}]=\{a+b\sqrt{15}:a,b\in\mathbb{Z}\}$. How do I show that $(3,\sqrt{15})$ is a maximal ideal but not a principal ideal? How do I show that $(3,\sqrt{15})^2$ is a ...
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1answer
29 views

Showing that $\operatorname{lcm}(a,b)$ is the unique positive generator of $(a) \cap (b)$

Let $a,b \in \mathbb{N}$. $l>0$ is the unique positive generator of the ideal $(a) \cap (b)$. Show that $l = \frac{ab}{d}$ where $d = gcd(a, b)$. I am stuck on this problem. $(a)=\{na: n \in ...
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1answer
65 views

Principal Ideals and Identity elements

When defining a principal ideal, e.g $I$ $I=aR=(ar:r \in R)$ (an ideal generated by a single element of the ring)do we require the ring to have an identity element and if so, in what manner does ...
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1answer
31 views

Uncountably many left ideals?

Let $R$ be a following subring of $M_2(\mathbb{C}):$ \begin{equation*} R = \left\{ \begin{bmatrix} a & r \\ 0 & s \end{bmatrix} ~:~ a\in \mathbb{Q} ...
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1answer
44 views

Determine if an ideal is prime

Let $R=\mathbb{Z}[x,y]$ and $I=(y-x^2)R+(x-4)R$ is ideal $J$ prime ? I tried to produce such $a,b$ that $ab \in J$ but $a,b \not\in J$ but can't find so far
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For ideals $I=(8)$ and $J=(5+5i)$ in $\mathbb{Z}[i]$, what are $IJ$, $I+J$ and $I\cap J$?

Let $I=(8)$ and $J=(5+5i)$ be ideals in $\mathbb{Z}[i]$. How do I find $x,y,z\in\mathbb{Z}[i]$ such that: $IJ=(x)$, $I+J=(y)$, $I\cap J=(z)$? Is it correct that $y=13+5i$ and $x=40+40i$?
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1answer
48 views

Prove that $\operatorname {ht}(p/a)\leq \operatorname {ht}(p)\leq \operatorname {ht}(p/a)+n$

In the Milne's book A Primer of Commutative Algebra, pg. 100, there's a proof that $\operatorname {ht}(p/a)\leq \operatorname {ht}(p)\leq \operatorname {ht}(p/a)+n$. I understand the first inequality, ...
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1answer
16 views

$A_i \subset$ ring $R$ st $\Delta A_i \equiv \{ b-c: b,c \in A_i\} = $ an ideal of $R$, then $\bigcap_i (\Delta A_i) = \Delta (\bigcap_i A_i)$.

Let $A$ be any subset of a ring $R$. $A$ is called a delta generating set of an ideal if $\Delta A = \{b - c: b,c \in A\}$ forms an ideal of $R$. Let $A_i$ be any collection of delta generating sets ...
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What does the ideal $I$ of $\mathbb{Z}[X]$ generated by $x$ and $2$ look like? [duplicate]

What does the ideal $I$ of $\mathbb{Z}[X]$ generated by $x$ and $2$ look like? I don't know how to put it into terms of more explicit set notation
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3answers
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What would an ideal I generated by e.g. 15 and 12 look like? What would the quotient ring $\mathbb{Z} / I$ look like?

What would an ideal I generated by e.g. 15 and 12 look like? What would the quotient ring $\mathbb{Z} / I$ look like? How do I find a formal representation for this quotient ring? Thanks
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1answer
42 views

Given a ring $R$, $R=(1)$ is a principal ideal

An ideal generated by the element $a$ is defined to be the intersection of all ideals containing $a$. My book says $R=(1)$ is a principal ideal, and I know how to convince myself of this using ...
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2answers
63 views

Correspondence between nilpotents and between idempotents

It is well-known and easily proved that whenever $R$ is a commutative ring with unity and $S$ is a multiplicative subset of $R$, each ideal of the localization ring $R_S$ is an extended ideal (with ...
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1answer
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Prove that $m_1m_2\ldots m_r=n_1n_2\ldots n_s$ implies $r=s$ for distinct maximal ideals

Let $R$ be a commutative ring where $m_1,m_2,\ldots,m_r$ and $n_1,n_2,\ldots,n_s$ are maximal ideals such that $m_1m_2\ldots m_r=n_1n_2\ldots n_s$ and $m_i \neq m_j$, $n_i \neq n_j$ if $i \neq j$. ...
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1answer
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Proving $R/J$ is local, where $R=k[\Gamma]$ and $J=(x^1)\unlhd R$.

Let $\Gamma$ be the set of symbols of the type $x^q$, where $q\in\Bbb Q, \;q\ge0$. Setting $x^{q_1}\cdot x^{q_2}:=x^{q_1+q_2}$, $(\Gamma,\cdot)$ becomes a semigroup. Let then $k$ be a field. Let's ...
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On publication regarding right ideals of a ring and the sublanguages of science [closed]

As some of you may know (or may experience by searching some of my threads), I have been working on the applications of right ideals of a ring to the study of language (in particular, to the so-called ...
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33 views

If $\phi:A\to B$ is a ring homomorphism, why does there exist $\psi:\text{spec}(A)\to \text{spec}(B)$?

Let $\phi:A\to B$ be a ring homomorphism, where $A$ and $B$ are commutative rings. We know that if $q$ is a prime ideal in $B$, then $\phi^{-1}(q)$ is a prime ideal in $A$. Hence, there exists a ...
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1answer
40 views

Factorization of ideal in field $\mathbb{Q}(\sqrt[3]{2})$ and its normal closure

So far I've worked only with quadratic fields, and I'm not sure how to work with 3rd roots. I have ideal $(5)$ and need to factor it in $\mathbb{Q}(\sqrt[3]{2})$ and its normal closure. I know that ...
3
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1answer
27 views

Extension of idempotent ideals

Let $R$ be a Noetherian commutative ring with $1$. If $R[[x]]$ denotes the ring of formal power series over $R$ and $I$ is an idempotent ideal of $R$ I want to know whether the extension of $I$ in ...
4
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2answers
92 views

Let $(R,M)$ be a local ring. Suppose that $R$ is noetherian and let $I,J \unlhd R$ such that $J \subseteq I$. Prove that the following are equivalent.

Let $R$ be a local ring with maximal ideal $M$. Suppose that $R$ is noetherian and let $I,J$ be ideals of $R$ such that $J \subseteq I$. Consider the following statements: 1) Every minimal set of ...
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2answers
33 views

Which of the sets are ideals and maximal ideals?

The exercise asks me to prove which of the sets are ideals, and if they are, which of those are maximal. I have these 4 cases: $$ a) J = \{f(x)\in \mathbb{Q}[x]: f(1)=f(7)=0 \} \\b) J = \{f(x)\in ...
2
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2answers
35 views

Proving $\mathbb{Q}[\sqrt{2}] = \{f(\sqrt{2}): f(x) \in \mathbb{Q}[x]\} = \{x+y\sqrt{2}:x,y\in\mathbb{Q}\}$

I need to prove that: $$\mathbb{Q}[\sqrt{2}] = \{f(\sqrt{2}): f(x) \in \mathbb{Q}[x]\} = \{x+y\sqrt{2}:x,y\in\mathbb{Q}\}$$ Well, $ \{f(\sqrt{2}): f(x) \in \mathbb{Q}[x]\} $ is the set of ...