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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Showing the sum of a C* subalgebra and ideal is itself a C* subalgebra

In my functional analysis class I was recently met with this in the context of C* algebras: Let A be a C*-Algebra and B is a C*-subalgebra of A and I an ideal of A. We are asked to show that $ B+I ...
3
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1answer
76 views

Exercise on radical ideal and formal derivatives

I need some help for solving the following exercise, because at the moment I'm a little bit lost and don't know where to start. Given a field $k$ with $\mathrm{char}(k)=0$ and a polynomial $f\in k[...
3
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2answers
32 views

Reduced ring are SI?

A ring $R$ is called an SI-ring if for any $a\in R$ the right annihilator of $a$ is an ideal of $R$. It is equivalent to the following statement: "if $ab=0$ for $a,b\in R$ then $aRb=0$". Is it true ...
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1answer
33 views

Listing all the ideals of a quotient ring [closed]

I have no idea how to answer this question. Let R be the quotient ring $\mathbb Q[X]/(X^3 + X^2 + X + 1)$. How to list all the ideals of R? And how to determine whether each ideal is prime, ...
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0answers
43 views

Show that there are finitely many different principal ideals [duplicate]

Let $R$ be a U.F.D. and $0\neq d\in R$. I want to show that there are finitely many different principal ideals that contain the ideal $(d)$. $$$$ We have that $R$ is a U.F.D. iff $\forall r\in R\...
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1answer
52 views

Show $Z(yf-1)$ is irreducible.

Question: $k$ is an algebraically closed field. Let $f \in k[x_1, \ldots, x_n]$ be an irreducible polynomial. Show that $Z(yf-1)\subseteq \textbf{A}^{n+1}$, with coordinates $x_1, \ldots, x_n, y$, ...
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3answers
70 views

Each prime ideal contains an idempotent element

An element of ring $e$ is called idempotent iff $e^2=e$. Let $R$ be a commutative ring that contains the identity element and a non-trivial idempotent element. I want to show that each of ...
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2answers
35 views

(0) is Maximal Ideal of Q(i) [closed]

Prove or disprove Ideal generated by $(0)$ is maximal ideal in $\mathbb{Q}(i)$.
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49 views

Can every polynomial generate an ideal?

Suppose an arbitrary polynomial $f$ in a polynomial ring $R$. Is $\langle f\rangle$ always an ideal? Helper parts Consider a finite polynomial ring. Let $R=R[x_1,\ldots,x_n]$. Is the answer ...
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0answers
31 views

How is “Binomial” defined in Algebraic Geometry?

I am learning ideal arithmetics and I was flabbergasted that $\langle x\rangle$ is binomial ideal, as observed with Macaulay2 here. $x$ is clearly not a polynomial with two terms. Then I read paper ...
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29 views

Binomial ideals that are not toric i.e. binomial and not prime?

Let $R$ be a ring. Toric ideal $I$ is binomial (generated by a binomial) and prime (the quetient ring $R/I$ is integral domain). Paper and corollary 1.3 is to determine whether an ideal is binomial. ...
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1answer
21 views

$R$ commutative with identity, $I$ finitely generated ideal. For each $r\in I$ there exists $m$ such that $r^mA\subseteq C$. Show $I^nA\subseteq C$.

I'm stuck on an exercise (E.VIII.4.1) from Algebra by Hungerford. It is stated as follows (sorry for the bad title; I didn't know how to fit it): Let $R$ be a commutative ring with identity and $I$...
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1answer
16 views

Is the difference of ideals an ideal?

I am studying ideals and noticed that $I+J$ is an ideal as noted here. However the paper does not discuss $I-J$ so: Is the difference of ideals an ideal?
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1answer
56 views

Has toric ideal something to do with torus?

I am studying ideals such as toric ideals but I am unable to find a consistent definition, it seems to be very general so please explain the origin of "toric ideal". Is there a geometric ...
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1answer
37 views

Each element is invertible [closed]

Let $R$ be a ring and let $I\subseteq R$ the only maximal right ideal of $R$. I want to show that each element $a\in R-I$ is invertible. $I$ is also an ideal. Could you give me some hints what I ...
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1answer
53 views

Algebraic number theory, Marcus, Chapter 3, Question 9

Question 9 in Marcus book. Let $K$ and $L$ be the number field such that $K\subset L$ and let $R,S$ be their algebraic integers, respectively. a) Let $I$ and $J$ be ideals in $R$, and suppose $IS|...
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2answers
71 views

How can we show that $I$ is an ideal? [duplicate]

Let $R$ be a ring and $I$ the set of non-invertible elements of $R$. If $(I,+)$ is an additive subgroup of $(R,+)$, then show that $I$ is an ideal of $R$ and so $R$ is local. $$$$ I have done ...
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3answers
41 views

Norm of element $\alpha$ equal to absolute norm of principal ideal $(\alpha)$

Let $K$ be a number field, $A$ its ring of integers, $N_{K / \mathbf{Q}}$ the usual field norm, and $N$ the absolute norm of the ideals in $A$. In some textbooks on algebraic number theory I have ...
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1answer
76 views

Show that $I$ is an ideal

Let $R$ be a ring and $I\subseteq R$ the only maximal right ideal of $R$. I want to show that $I$ is an ideal. To show that $I$ is an ideal, we have to show that $I$ is a left ideal, right? How ...
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0answers
33 views

$(2, 1+\sqrt{-5})$ is a prime ideal in $\mathbb{Z}[\sqrt{-5}]$ [duplicate]

I need a hint (just a hint please, not a full answer) to proving that $(2, 1+\sqrt{-5})$ is a prime ideal in $\mathbb{Z}[\sqrt{-5}]$. I'm trying to prove it via definition of a prime ideal and ...
1
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1answer
42 views

Show that $I^{(n)}$ is a primary ideal belonging to $P$ [duplicate]

Let $A$ be a commutative ring with identity, $P$ a proper prime ideal in $A$, $I$ a primary ideal belonging to $P$ and $n$ a positive integer. The ideal $(I^{n})^{ec}$ (extension and contraction being ...
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1answer
27 views

Let $A$ be a ring and $m_1,…,m_k$ maximal ideals

Let $A$ be a ring and $m_1,...,m_k$ maximal ideals of $A$, not necessarily different, and $F_i=m_1\cdots m_{i-1}/m_1\cdots m_i$. Because $m_iF_i=0$, $F_i$ can be made into a $A/m_i$-module defining ...
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41 views

Proof of commutative Artinian ring is Noetherian

I think that I have a proof, but it seems much simpler than all proofs that I can find on the internet. Hence I suppose that there must be a mistake in my proof. The commutative ring $R$ is ...
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1answer
24 views

Key reference book on toric ideals: normal or not? Which definition to follow?

I want to understand sum of binomials better in terms of ideals such as binomial ideals, normal ideals and so by toric ideals. Examples about toric ideals contain $$\sum x^\alpha+\sum x^\beta\in\...
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1answer
26 views

Is there a non-trivial ordered ring with an “integer-esque” modulo function?

(I'm inspired by this question.) Is there a [not-necessarily-commutative non-simple ordered ring with a 1 that's not equal to 0] which is not isomorphic to the integers but is such that for all ...
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0answers
33 views

Ideal generated by a set of polynomials $X^{a/b}$ where each monomial having $a$ and not having $b$

Let $$\mathcal R=\mathbb Z_2[x_1,\dots,x_n]/\langle x_1^2-x_1,\dots,x_n^2-x_n\rangle.$$ I want to learn ideal arithmetics to deal with polynomials of the forms such as Consider a set of ...
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1answer
29 views

What is this ideal equal to? What is it called? “composition ideal in $R[X]$”

Let $R$ be a ring and $f(X)=f_0+f_1X+\dots +f_n X^n\in R[X]$. Define $f(J) \equiv f_0 + f_1 J + \dots + f_n J^n$ where $J^k$ is the $k$th power ideal, and $A + B = \{a + b : a \in A, b \in B\}$. ...
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2answers
42 views

Showing that $\langle x,y\rangle^2 \subsetneq \langle x,y^2\rangle$ [closed]

I've the ring $K[x,y]$, where $K$ is a field. How should I show that the ideal $\mathfrak q=\langle x,y^2\rangle$ contains ideal $\mathfrak p^2$ properly, where $\mathfrak p=\langle x,y\rangle$. I ...
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1answer
76 views

Assume that $I/J$ is a prime ideal of $R/J$, is $I$ a prime ideal of $R$? [closed]

$R$ is a commutative ring. $I$ and $J$ are ideals of $R$ with $J\subseteq I$. Assume that in the quotient ring $R/J$, $I/J$ is a prime ideal. Is the ideal $I$ a prime ideal in $R$?
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If $x$ is not nilpotent, how to prove there exists a prime ideal does't contain $x$

Actually, I find some explanation using Zorn's lemma and localization. However, our class doesn't include these until now. So can someone prove it in an easier way?
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Show pR[x] + (x) is a prime ideal.

I am self studying the notes here. The problem is exercise 2.18 on page 9 (solutions provided there as well). Let R be a ring, p a prime ideal, R[X] the polynomial ring, pR[x] the product ideal ...
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1answer
30 views

Sum of nil right ideals

Is an arbitrary sum of nil (nilpotent) right ideals a nil (nilpotent) right ideal? If $I=\sum I_i$ is a sum of nil ideals then each element $x$ of $I$ is a finite sum $x=x_1+...+x_n$ of elements $x_k\...
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1answer
23 views

Proving the Nilradical of a Commutative Ring is closed under addition. [duplicate]

Let $R$ denote a commutative ring and $Nil(R)$ is the ideal consisting of all nilpotent elements in $R$. I am attempting to prove that $Nil(R)$ is closed under addition. My work so far is ...
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1answer
104 views

Show that a radical ideal has no embedded prime ideals. [closed]

Let $A$ be a commutative ring and $I$ a decomposable ideal. Let $I=\bigcap_{k=1}^{n} I_k$ be a minimal primary decomposition. Show that if $I=\sqrt{I}$ then $I$ has no embedded prime ideals. (I ...
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1answer
46 views

polynomials modulo even numbers

Say I have $R= \mathbb{Z}[x]$ and $A = \{p_0+p_1x+p_2x^2+\cdots+p_nx^n \mid n\geqslant0, p_i\in\mathbb{Z}, p_0, p_1 \text{ even}\}$. Define $K=R/A$. How would I characterize the elements of $K$? ...
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38 views

An irreducible variety is not composed of finitely many subvarieties

There is a lemma in commutative algebra: Let $\mathfrak{a}_1, \dotsc, \mathfrak{a}_n$ be ideals such that $\mathfrak{a}_n \cap \dotsb \cap \mathfrak{a}_n$ is contained in a prime ideal $\...
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0answers
43 views

Is $x$ a member of an ideal? Is $x$ a member of the radical? [duplicate]

Consider the ideal $ J := \langle x^3y-x^2y^2,x^3z+z^2yx,x^2-xz\rangle\subset \Bbb{Q}[x,y,z]$. Is $x\in J$? Is $x \in \sqrt J$ ? I'm not sure how I go about showing these two questions, can anyone ...
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Show $\mathbb {R}[x,y]/(y^2-x, y-x)$ is not an integral domain

Let $\mathbb{R}[x,y]$ denote the polynomial ring in two variables $x$, $y$ over $\mathbb{R}$, and let $I = (y^2-x,y-x)$ be the ideal generated by $y^2-x$ and $y=x$. Show that $$\mathbb{R}[x,y]/I$$ is ...
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1answer
68 views

Noetherian rings have only finitely many minimal prime ideals. [duplicate]

We say that $p$ is minimal prime if It does not contain any other prime. Assume that $A$ is Noetherian ring Question: $A$ has only finitely many minimal primes. any suggestions please
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1answer
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Jacobson radical of polynomial quotient ring.

Let $F$ be a field, and $A=F[x]/(x(x-1)^2)$. 1. Find the ideals of $A$. Which of them are simple or maximal? 2. Find the Jacobson radical, $J(A)$, of $A$. 3. Find two composition series for $A$, as ...
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20 views

Are all integral domains in which all irreducible elements are prime G.C.D domains?

I know that in G.C.D domains all irreducible elements are prime. Does the converse of this statement hold? If not, is there a weaker condition than being a G.C.D. domain that is both sufficient and ...
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2answers
79 views

An example of ideal $I$ such that $I^{ec}\neq I$ [on hold]

Let $A$ be a commutative ring, $S \subseteq A$ a multiplicative system and $i_S : A \rightarrow S^{-1}A$ the canonical morphism. Can you give me an example of ideal $I \unlhd A$ such that $I^{ec}\neq ...
2
votes
2answers
70 views

How is finiteness of solutions (zero-dimensionality) related to Krull's dimension?

I have encountered a lot the concept of zero-dimensional ideal: Let $k$ be a field. An ideal $I\subseteq k[x_1,...,x_m]$ is said to be zero-dimensional if its zero set $Z(I)$ has a finite number ...
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Does UFD $R$ wrap around its ideal $I$ infinitely and surjectively as $\Bbb{Z}$ does $(n)$?

If $R$ is a UFD, and $I$ is an ideal of $R$, then do elements of $R$ wrap around $I$ as they do in the case of $\Bbb{Z}$ and $(n)$. And by that I mean, letting $\pi : R \to R / I$ be the natural ...
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1answer
162 views

Is every ideal of $R$ a sum of a nilpotent ideal and an idempotent ideal?

I have a question concerning the following local ring: $$R=K[X_1,...X_n,...]/(X_1,X_2^2-X_1,...,X^2_{n+1}-X_n,...).$$ Is every ideal of $R$ a sum of a nilpotent ideal and an idempotent ideal? ...
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1answer
92 views

An example from Lang's Algebra about primary ideal

On page 421 in Lang's Algebra, the author writes Let $R$ be a factorial ring with a prime element $t$. Let $A$ be the subring of polynomials $f(X)∈R[X]$ such that $$f(X)=a_0 + a_1X + \dotsb $$ ...
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votes
1answer
99 views

Prove that $\langle\sqrt2\rangle$ is a maximal ideal in $\Bbb Z[\sqrt2]$. How many elements are in $\Bbb Z[\sqrt2]/\langle\sqrt2\rangle$? [closed]

Prove that $\langle\sqrt{2}\rangle$ is a maximal ideal in $\Bbb Z[\sqrt{2}]$. How many elements are in the ring $\Bbb Z[\sqrt{2}]/\langle\sqrt{2}\rangle$ ? I am unable to solve this. Please help me ...
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2answers
36 views

How to solve this problem about faithful module.

Let $R$ be a non-zero commutative ring with identity and $M$ a unital $R$-module. The $R$-module $M$ is called faithful if $rM=0$ for $r\in R$ implies $r=0$. Let $M$ be a finitely generated ...
0
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2answers
67 views

Radical of the powers of an ideal

I am asked to prove the following: $$\sqrt{\mathfrak{a}^n} = \sqrt{\mathfrak{a}}$$ Here is my attempt so far: $\sqrt{\mathfrak{a}^n} \subseteq \sqrt{\mathfrak{a}}:$ (By Induction) Clearly the ...
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2answers
45 views

Sum of Ideals of the Same Type

I have two questions: 1) Is a finite sum of idempotent ideals of a ring $R$ idempotent? 2) Is any sum of nil ideals of a ring $R$ nil? As far as I know, a finite sum of nil ideals of a commutative ...