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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About ring principal ideal [duplicate]

Have this homework assigment, able to do it, except last part. Here is exercise: 1.Prove that set $I=\{f(x)\in\mathbb{Z}[x]|f(0) \quad is\quad even\}$ is rings $\mathbb{Z}[x]$ ideal 2.Prove that if ...
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24 views

Understanding Quotienting by Relations vs Quotienting by Generators

I understand the idea of a quotient algebra $A / I$ where $A$ is a $K$-algebra and $I$ is a two-sided ideal, i.e. I understand the projection map as an algebra morphism. However, I'm unsure about how ...
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Applications of one-sided ideals in programming, search or processing

As many of you may already know, I am particularly interested in one-sided ideals in a ring (mostly, right ideals). The thing is, I would like to know about applications in the realm of computer ...
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How do I show this assertion? [duplicate]

Show that the ideal generated by $x^2-y$ is a prime ideal in $C[x,y]$. It would be sufficient if we show that $C[x,y]/<x^2-y>$ is an Integral Domain. Or is there any other way of showing the ...
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23 views

Show that the set of all polynomials $f$ in $F[x]$ such that $f(A)=0$ is an ideal

Let $A$ be an $n \times n$ matrix over a field $F$. Show that the set of all polynomials $f$ in $F[x]$ such that $f(A)=0$ is an ideal. I don't understand how to apply this when it comes to ...
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26 views

Let $R$ be a ring. Let $I\lhd R$ and fix $n\in I$ if $n$ is the unit of $R$. Show $R=I$

Let $R$ be a ring. Let $I\lhd R$ (that is $I$ is an ideal of the ring) and fix $n\in I$ if $n$ is the unit of $R$. Show $R=I$. Here is my attempt at an answer: We aim to show $I \subseteq R$ and $R ...
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39 views

Manually computing ideal quotient $\langle x\rangle : \langle x y z \rangle$ in $k[x,y,z,o]$

Please explain this ideal quotient in $k[x,y,z,o]$: $$\langle x\rangle : \langle x y z \rangle=\{f\in k[x,y,z,o] : fg\in \langle x \rangle\quad\forall g\in \langle x y z \rangle \}$$ where ...
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28 views

Let $d \in \mathbb{Z}$, $d > 1$. Determine all the ideals of $\mathbb{Z}/d\mathbb{Z}$ which are prime or maximal

Let $d \in \mathbb{Z}$, $d > 1$. Determine all the ideals of $\mathbb{Z}/d\mathbb{Z}$ which are prime or maximal I know that $m\mathbb{Z}/md\mathbb{Z} \cong \mathbb{Z}/d\mathbb{Z}$ as rings ...
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38 views

P.I.D. and a nontrivial ideal, Quotient ring has finitely many ideals [on hold]

A ring $R$ is a P.I.D. Let $I$ be a nontrivial ideal in $R$. Prove that $R/I$ has finitely many ideals.
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Affine varieties and their ideals (part2)

On wikipedia, they talk about varieties $V,W$ and the $I(V)$ and $I(W)$ as well as the quotient ideal, $$I(V):I(W) = I(V - W)$$ Can someone show me a quick proof of the identity?
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47 views

A subset of a polynomial ring and its ideal. [duplicate]

Let $P=K[x_1,\dots,x_n]$ be a polynomial ring over a field $K$ and $I = (f)$ be a principal ideal in $P$ generated by $f \in P - \{0 \}$. Moreover let $L \subset \{x_1, \dots, x_n \}$ and $\hat{P} ...
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24 views

A $R$-module over a ring $R$ with identity is free iff it is a direct sum of copies of $R_R$, where $R_R$ is $R$ considered as a module over itself

Let $R$ be a ring with identity and $M$ be an $R$-right module. Then $M$ is called free over $X \subseteq M$ if for every module $N$ with mapping $\alpha : X \to N$ we can extend it uniquely to a ...
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71 views

Show that the only nonzero ideals of R are the principal ideals $\langle p^e \rangle$

Let $p$ be a prime number in $\mathbb{Z}$. Let $R = R_p = \{x \in \mathbb{Q}\ |\ \textrm{ord}_p(x)\geq0\}$, which is a subring of $\mathbb{Q}$. Show that the only nonzero ideals of $R$ are the ...
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Use Gröbner bases to count the $3$-edge colorings of planar cubic graphs…

I found a nice introduction on how to Use Gröbner bases to construct the colorings of a finite graph. Now my graphs $G=(V,E)$ are the line graphs planar cubic graphs, so they are $4$-regular. The ...
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Representation of left ideals of the matrix ring [duplicate]

We know that $J$ is an ideal of the full matrix ring $S=M_n(R)$ if and only if $J$ is the ring of all $n\times n$ matrices over $I$ for some ideal $I$ of the ring $R$ with identity. Now, my question ...
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Show that the set of all principal ideals is an equivalence class of the relation $\sim$

Let $A$ a integral domain and let $\mho(A)$ the set of all non-zero ideals. Show that the set of all principal ideals is an equivalence class of the relation $\sim$ that we can noted by $[A]$. ...
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13 views

Kernel of $M\to M[U^{-1}]$ and primary decomposition of $(0)$

I am working on exercise 3.12 from Eisenbud's Commutative Algebra and I am having trouble parsing the question. Let $M$ be a finitely generated module over the Noetherian ring $R$. Given any ...
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1answer
33 views

If I is an irreducible ideal, and P is a prime ideal, is (I+P)/P irreducible?

Let $A$ be a commutative ring with unit, and $P$ a prime ideal. My question is: If $I$ is an irreducible ideal in $A$, is $(I+P)/P$ irreducible in $A/P$? If not, can you show a counterexample? ...
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$m_p=\{f\in \mathcal{O}_{V,p}| f(p)=0\}$, ideal of $p$ in the local ring. What is $m_p/m_p^2$?

In Section 6.8 of Undergraduate Algebraic Geometry by Reid, the author proved the following Theorem: There is a natural isomorphism of vector spaces $(T_pV)^*\cong m_p/m_p^2$ where $^*$ denotes ...
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49 views

Correspondence between prime and maximal ideals [closed]

My professor put the following statement in the lecture notes without proof: Let $R$ be a commutative ring and $I$ an ideal. Then the natural correspondence between ideals containing $I$ and ideals ...
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44 views

Intersection of two ideals

Let $A$ be a commutative ring and let $\mathfrak{a}$, $\mathfrak{b}$ be ideals in $A$. I am asked the following question: Show that $\mathfrak{a} \cap \mathfrak{b}$ is the largest ideal of $A$ ...
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28 views

Ideals and submodules are the same [closed]

My teacher has told me that for an R-module, that I is an ideal of R if and only if I is an R-submodule of R. I know this is true but I was wondering why? IS there an official proof?
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Ideals of $\mathfrak{gl}_n$

How does one determine the ideals of $\mathfrak{gl}_n(C)$? My guess is that the only ones are $(0) $ and $\mathfrak{sl}_n(C)$. I think approaching the problem by the fact that each $\mathfrak{g}^{ ...
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Is the set of all $f$ such that $\lim_{x\to1^-}f(x) = 0$ an ideal in the ring of functions from $[0,1]\rightarrow \mathbb{R}$?

Is the set of all $f$ such that $\lim_{x\to1^-}f(x) = 0$ an ideal in the ring of functions from $[0,1]\rightarrow \mathbb{R}$? I'm sure about the closure under addition but not quite clear about if ...
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21 views

Union Over a Totally Ordered Set of Ideals is an Ideal

I am trying to understand the proof of a theorem which uses Zorn's lemma. I understand quite well all parts of the proof except for one point: Let $R$ be a ring and define $K\doteq \{I\subseteq R ...
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73 views

When is an ideal also a ring, and could then be anything said about its relation to the original ring

If $R$ is a ring with unity $1$, then $S \subseteq R$ is called a subring if it is itself a ring with $1 \in S$. A subset $I \subseteq R$ is called an ideal if it is a group with respect to addition ...
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Let A and B be ideals of a ring and C a prime ideal. Prove if the intersection of A and B is a subset of C then either A or B is a subset of C

Claim: Let $A$ and $B$ be ideals of a commutative ring $R$ and $C$ a prime ideal of $R$. Suppose that the intersection of $A$ and $B$ is a subset of $C$. Prove either $A$ or $B$ is a subset of $C$. ...
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20 views

Show that the left ideal $(N_G) \subset F[G]$ is a simple submodule of $F[G]$, where $N_G = {\sum}_{g \in G} {g} \in F[G]$. [duplicate]

I am trying to solve this Representation Theory question: Let $F$ be a field and $G$ a finite group. Let $N_G = {\sum}_{g \in G} {g} \in F[G]$. Show that the left ideal $(N_G) \subset F[G]$ is a ...
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36 views

Show that the ideal generated by $x^2-2$ is maximal

Let $A = \mathbb{Q}[x]$. Show that the ideal generated by $x^2-2$ is maximal. I think it is sufficient to show that $\mathbb{Q}[x]/(x^2-2) \cong \mathbb{Q}\sqrt{2}$, where $\mathbb{Q}\sqrt{2} = \{a + ...
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Is the ideal generated by the polynomial $x^2-2$ maximal in the ring $\mathbb{C}[x]$?

Personal question: Is the ideal generated by the polynomial $x^2-2$ maximal in the ring $\mathbb{C}[x]$? I know the ideal in $\mathbb{Q}[x]$ generated by $x^2-2$ is maximal, considering ...
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Groebner basis and prime ideals.

Let $I$ be an ideal in a polynomial ring $P = K[x,y_1,\dots,y_n]$ and assume that $I \cap K[x]\neq (0)$. Let $>$ be an elimination ordering for $\{y_1, \dots, y_n\}$ and $G$ is a Groebner basis for ...
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Why is $|V(I)| \leq d_1\cdots d_n$?

If $I \subset K[x_1,\dots,x_n]$ is a zero dimensional ideal and $$V(I) = \{ (\alpha_1,\dots,\alpha_n) \in K^n: f((\alpha_1,\dots,\alpha_n)) = 0\ \forall f\in I\}$$ (the variety). Then if $G$ is a ...
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Minimal primary decomposition of the ideal $I = (XY, Y Z, XZ) ⊆ \mathbb C[X, Y, Z]$ [duplicate]

Write out a minimal primary decomposition of the ideal $I = (XY, Y Z, XZ) ⊆ \mathbb C[X, Y, Z]$, and determine the primes belonging to $I$. Determine the dimension of the ring $\mathbb C[X, Y, ...
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Find an infinite collection of maximal ideals containing $(x^2 - y^3) \subset \mathbb{C}[x,y]$ [closed]

What is an infinite collection of maximal ideals containing the ideal $I = (x^2 - y^3) \subset \mathbb{C}[x,y]$?
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1answer
35 views

Quotient of maximal and prime ideals [closed]

Given that $I, J$ are ideals in $R$, $I$ is maximal or prime, do we have that $I/J$ is maximal in $R/J$? $I/J$ is prime in $R/J$? I think it is true but don't see how it works.
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Do there exist polynomials $f,g$ in $\mathbb{C}[x]$ such that $(x^2 - 1)f + x = g^2$.

Do there exist two polynomials $f, g \in \mathbb{C}[x]$ such that $(x^2 - 1)f + x = g^2$? I know that this cannot happen in $\mathbb{R}[x]$. However, since $\mathbb{C}$ is algebraically closed, this ...
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Principal Ideal Domain $R$ and ideal $J\neq 0$ so that that $R/J$ have a finite number of ideals. [duplicate]

Let $R$ a un Principal Ideal Domain(PID) and $J\neq 0$ a ideal of $R$. Show that $R/J$ have a finite number of ideals.
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When is the ideal generated by 2 elements equal to the sum of the 2 ideals

Is it true in general that (a,b)=(a)+(b)? I would suppose that (a)+(b)$\subset$(a,b) and i believe the reverse containment should hold as well, i just can't seem to fit the pieces together.
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Laurent Ideal whose Intersection with Polynomial Ring Requires More Generators

I want to find an ideal $I\subseteq \mathbb Q[x^{\pm 1}, y^{\pm 1}, z^{\pm 1}]$ which requires fewer generators than the affine ideal $I\cap \mathbb Q[x, y, z]$. I tried finding a principal ideal $I$ ...
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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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29 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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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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1answer
44 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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40 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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22 views

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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1answer
27 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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23 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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88 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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1answer
51 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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33 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 ...