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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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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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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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 ...
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25 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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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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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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Showing that $\langle x,y\rangle^2 \subsetneq \langle x,y^2\rangle$

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 $\mathfrak p=\langle x,y\rangle$. ...
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73 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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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 ...
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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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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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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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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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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 $$\mathbb{R}[x,y]/I$$ ...
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58 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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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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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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An example of ideal $I$ such that $I^{ec}\neq I$

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 ...
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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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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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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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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 ...
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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 ...
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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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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 ...
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50 views

Show that the Group Ring $F_p[G]$ where $G$ is a $p$-Group has a unique maximal ideal.

Show that the Group Ring $F_p[G]$ where $F_p$ is finite field of order $p$ and $G$ is a $p$-Group (not necessarily abelian) has a unique maximal ideal, i.e. it is a local ring. Attempt: Consider ...
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Contraction of an ideal

Let $f: \mathbb{Z}[X] \longrightarrow \mathbb{Z}[\sqrt{2}]$ be a ring homomorphism sending $X$ to $\sqrt{2}$. I am asked to compute a few contractions, and I am wondering if I could get some help ...
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Existence of minimal prime ideal contained in given prime ideal and containing a given subset

Let $R$ be a unital commutative ring, $P$ $\subseteq$ $R$ a prime ideal, $X\subseteq P$ a subset. Show there exists a minimal (inclusion minimal) prime ideal contained in $P$ which contains $X$. ...
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Understanding the concept of polynomial ideals

I am not able to understand the fundamental concept behind a polynomial ideal. What I have so far in terms of $I$ being an ideal of a ring is: for each $f, g \in I$, we have $-f$ and $f+g \in ...
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Ideal proof in a ring R

Problem Statement: Let I be an ideal in a ring R. Prove that K is an ideal, where $ K $ = { $a\in R$ | $ (\forall r\in R)(ra\in I) $} What exactly am I supposed to show here? I know I need to show ...
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71 views

Let $I$ and $J$ be ideals in $R$. Is the set $K= \{ ab \ | \ a\in I, b\in J \}$ an ideal in R? [duplicate]

I've just assumed that this is false, since the problem statements says to compare it to a previous problem where $\{ a+b \ | \ a\in I, b\in J \}$ is ideal. However, by trial and error I can't find ...
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Radical of an Ideal Proof

$\sqrt{\sqrt{\mathfrak{a}}} = \sqrt{\mathfrak{a}}$ $\sqrt{\sqrt{\mathfrak{a}}} \subseteq \sqrt{\mathfrak{a}}:$ Let $x \in \sqrt{\sqrt{\mathfrak{a}}}$, then $x^n \in \sqrt{\mathfrak{a}}$ for ...
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radical membership and ideal membership [closed]

Consider the ideal $I=(x^3y-x^2y^2,x^3z+z^2yx,x^2-xz)\subset \Bbb Q[x,y,z].$ Is $x\in I?$ Is $x\in \sqrt I?$ I'm assuming a question like this is quite simple and that there is just a method, if ...
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Prove that an ideal quotient is an ideal

Let R be a commutative ring with identity and I, J be ideals in R. $I:J$ = {$r∈R|rj∈I, \forall j∈J$} Prove that $I:J$ is an ideal in $R$, and contains $I$. To begin this, I wrote $I:J$ in the form ...
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Ring morphism, find the kernel

I am trying to solve the following problem Let $k$ be a field and consider the ring morphism $f:k[x,y] \to k[t]$ defined by $f(x)=t$ and $f(y)=q(t)$ with $q(t) \in k[t]$. Find the kernel of $f$. ...
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A necessary and sufficient condition for determining prime ideals in any ring?

Definitions Definition 1. A set $S$ equipped with two binary operations '$+$' and '$\cdot$' will be said to be a pseudo-ring if, $(S,+)$ is a monoid $(S,\cdot)$ is a semi-group $a(b+c)=ab+ac$ and ...
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35 views

Radical ideals in $\mathbb Z[x]$ such that their sum is not radical

I am trying to solve an exercise in which I have to provide an example of two radical ideals $I,J \subset \mathbb Z[x]$ such that their sum $I+J$ is not radical. I don't know how to attack this ...
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If $R$ is a domain and $M_n(R)$ is semisimple, then $R$ is a division ring.

From Lam's A First Course in Noncommutative Rings, section 1.3. Let $R$ be a domain (EA: that is, a ring without zero divisors) such that $M_n(R)$ is semisimple. Show that $R$ is a division ring. ...
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36 views

The factor ring of the n-th power of a maximal ideal is local. [closed]

Let $M$ be a maximal ideal in a commutative ring $R$ with identity and $n$ is a positive integer, then the ring $R/M^n$ has a unique prime ideal and therefore is local. It is easy to see that unique ...
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Why is the generating set a proper ideal of…? [closed]

Why is $\langle 89, 3-4\sqrt{-5}\rangle$ a proper ideal of $\Bbb{Z}[\sqrt{-5}]$?
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Polynomial Ring Divided By Principal Ideal

Let $F_5[X]$ be the polynomial ring over $F_5$ and $I = <X^2+X+1> $. Show that any element of $\frac{F_5[X]}{I}$ can be written as $a+bX+I$ where $a,b$ are in $F_5$. I guess I can be written ...
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Cluesless over a proper ideal question [duplicate]

Let $X=\{ \alpha+\beta\sqrt{-5} \mid \alpha,\beta \text{ are rational numbers} \}$ Does there exist an integer $a$ such that $a$ and $3-4\sqrt{-5}$ generate a proper ideal of $X$? Can anyone answer ...
2
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61 views

Number of generators of a given ideal.

Let $I=\langle 3x+y, 4x+y \rangle \subset \Bbb{R}[x,y]$. Can $I$ be generated by a single polynomial? My approach: If $I$ can be generated by a single polynomial, then the two "apparent" ...
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1answer
23 views

Showing that $1$ is an element of the ideal $\langle 3,x,n\rangle$ in $\mathbb Z[x]$ if $\operatorname{gcd}(3, n) = 1$ [closed]

Can anyone explain me the following step: If I have the ideal $\langle 3,x,n \rangle$ in $\mathbb Z[x]$ where $\operatorname{gcd}(3,n)=1$, then $1\in \langle 3,x,n \rangle$. Kindly help as I'm ...
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1answer
59 views

How to prove $(2)=(2x)$ for $x\in \mathbb{Z}$?

How to prove $(2)=(2x)$ for $x\in \mathbb{Z}$? ($(2)$ is the principal ideal generated by $2$ over the ring of integers.) I wrote $(2x)=\{(2x)z:z \in \mathbb{Z}\}=\{2y:y \in \mathbb{Z}\}=(2)$ where ...
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42 views

Isomorphism of a Quotient Ring

Show that $(A \times B) \big/ (\mathfrak{a} \times \mathfrak{b}) \cong \big(A \big/ \mathfrak{a} \big) \times \big( B \big/ \mathfrak{b} \big)$. Where $A$ and $B$ are commutative rings. I am ...