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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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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61 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 in ...
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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 $X$. My ...
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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 $f+g \in I$...
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30 views

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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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 some $...
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198 views

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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57 views

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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20 views

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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38 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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36 views

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 ...
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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" generators, ...
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25 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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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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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 asked ...
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54 views

Show that for a field $F$, the polynomial ring $F[x_1, x_2, \ldots, x_n]$ is not a PID for $n>1$.

I want clarification of the following solution: Let $I=(x_1)+(x_2)$ be an ideal of $F[x_1, x_2, \ldots, x_n]$. Then if $I=(f)$ is principal then we must have $f \in F \backslash \{0\}$ since $\gcd(...
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24 views

Determine a parameter so that a given ideal equals $ℝ[x,y]$

Let $$B=\langle 3x+y-m, 4x+y\rangle⊆ ℝ[x,y].$$ Find a parameter $m$ so that $$B=ℝ[x,y].$$ My attempt so far: If $1_{ℝ[x,y]}\in B$ then we get the desired equivalence, since any ideal ...
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Proving maximal ideals when considering an irreducible polynomial

For $F$ a field, and $q(x)\in F[x]$ Suppose that $q(x)$ is a irreducible polynomial within the ring. Prove that $ \langle q(x) \rangle$ is a maximal ideal of $F[x]$ I've already proved that $F[x]$...
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How to prove that two principal ideals are equal [closed]

Background info Provided that the general form of a polynomial is $(a_0X^{0}+...+a_nX^{n})$ where $X$ is an element of the field provided. an ideal generated by an element is the set $(a*r s.t. a\...
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Under what conditions is $J\cdot M$ an $R$-submodule of $M$?

I have that $M$ is an $R$-module where $R$ is commutative and unitary ring. Supposing that $J$ is an ideal of $R$, when is the set $J \cdot M$ an $R$-submodule of $M$? I have to check the two axioms ...
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58 views

Why this ideal is a subset of $K[x_0, \dots, x_n]$

For a point in the projective space $\mathbb{P}^n$, $p = [a_0:a_1:\dots :a_n]$, how to see that $$I(p) = \left < x_ia_j - x_ja_i: 0 \leq i \leq n, 0 \leq i \leq n \right> \subseteq K[x_0, \dots,...
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For ideal $m$ maximal and principal, there's no ideal between $m^2$ and $m$. Prove that this can be false when $m$ is not principal or maximal.

Prove that for ideal $m$ maximal and principal, there's no ideal $I$ such that $m^2 \subsetneq I \subsetneq m$. Show that this can be false when $m$ is not principal or maximal. Suppose $\...
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Find all maximal ideals of the ring $\mathbb Z_4 \oplus \mathbb Z_{15}$

Find all the maximal ideals of the ring $\mathbb Z_4 \oplus \mathbb Z_{15}$. The maximal ideal should be of the form $<1> \oplus <p>$ or $<p> \oplus <1>$ where $p$ is a ...
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The intersection and sum of irrelevant ideals are also irrelevant

Definition: A homogenous ideal $I \subset K[x_0,\dots,x_n]$ is irrelevant if $\left <x_0^r,\dots,x_n^r \right> \subset I$ for some $r > 0$. For $I \cap J$, this is probably circular logic, ...
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Describing integral ideals [closed]

Suppose I have a field $K=\mathbb{Q}(\sqrt{-d})$. How does one describe it's integral ideals?
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Prove that $J \cap R(1 - e) \neq \{0\}$

Let $R$ be a commutative unitary ring and I and J be two maximal ideals of $R$ such that $I \neq J$ and $I \cap J = Re$ where $e \in R$ is idempotent. Prove that $J \cap R(1 - e) \neq \{0\}$. I ...
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Verification of a factorization of ideals, $\langle 2 \rangle$

Still going over Alaca & Williams (I might die before I fully understand that book). In $\mathbb{Z}[\sqrt{-21}]$, the factorization of $\langle 2 \rangle$ is $\langle 2, 1 + \sqrt{-21} \rangle^2$....
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27 views

Chains of ideals contained in maximal ideal in non-Noetherian commutative ring

Given a maximal ideal $M$ in a non-Noetherian commutative ring $R$, I'm trying to determine whether or not there can exist infinite strictly ascending chains of ideals of $R$ contained in $M$. I know ...
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How to prove A is a flat R-module [duplicate]

Let A be a right R-module. Suppose for every left ideal J of R, the homomorphism $f:A\otimes J\to A$ defined by $f(x\otimes y)=xy$ is injective, then A is a flat R-module.(the identity 1 is in R) I ...
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Understanding properties of quotient rings

We have begun learning quotient rings in my Algebra course, but I am still confused by some of the theorems and properties of quotient rings. We have a theorem: $R$ is a ring, and $I\subset R$ ...
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Ideal of the union of two skew lines in $\mathbb{P}^{3}$

Let $$ L_{1}=V(X_{0},X_{1})\subseteq\mathbb{P}^{3}, $$ $$ L_{2}=V(X_{2},X_{3})\subseteq\mathbb{P}^{3}. $$ I want to prove that $$ I(L_{1}\cup L_{2})=(X_{0}X_{2},X_{0}X_{3},X_{1}X_{2},X_{1}X_{3}). $$ ...
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1answer
227 views

Is it true that if some power of an ideal is primary, then the ideal itself is also primary?

Is it true that if some power of an ideal $I$ is primary, then $I$ itself is also a primary ideal? I do not know whether the above statement is true or there is a counterexample. If one wants to ...
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What's so special about a prime ideal?

An ideal is defined something like follows: Let $R$ be a ring, and $J$ an ideal in $R$. For all $a\in R$ and $b\in J$, $ab\in J$ and $ba\in J$. Now, $J$ would be considered a prime ideal if ...
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1answer
49 views

Contraction and extension of ideals respect inclusions, sums and intersections

Let $R$ be an integral domain. Let $Y$ be a multiplicatively closed subset of $R$ which contains $1$ but not $0$. Define $S=RY^{-1}=\lbrace ry^{-1} : r \in R, y \in Y \rbrace$ as well as $\mathcal{I}(...
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1answer
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prove that maximal ideal in Z generated by a prime number

I am trying to prove <a> is a maximal ideal in Z, iff a is prime number. Now I wrote: assume a ∈ Z, while it's not prime number we can write as a=xy, for some integer x and y. then <a>⊂<x> and<a>⊂<y>...
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130 views

Is $F[x]/\langle x^3-1\rangle$ a field?

Let F be a field. Is $F[x]/\langle x^3-1\rangle$ a field? I know $x^3-1 = (x-1)(x^2+x+1)$. How can I use this to show the above is a field or not? I am using the above to deduce $\langle x^3-1\...
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73 views

Intersection of n hyperplanes in projective space of dimension n is not empty

I want to prove the following: Let $H_1,\dots,H_n$ be $n$ hyperplanes in $\mathbb{P}^n =\mathbb{P}^n \mathbb{C}$. Then $\cap_{i=1}^n H_i$ is not empty. Please be noted that this is an exercise ...
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Showing that an element generates the kernel

$I $ is a monomial ideal generated by $\left < m_1, \dots, m_n\right >$ and suppose we also have an $R$-module homomorphism $\phi: \oplus_{j = 1}^n Re_j \to I$ defined by $$\phi(e_i) = m_i.$$ ...
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33 views

Intersection multiplicity for two surfaces definded by $f=0,g=0$

I want to understand how I can find the intersection multiplicity $I_p$ at a point $p$ for two curves $f,g$. I have the example where $$ f(x,y) = y^2-x^3, \,\,\,\, g(x,y)=y^2-x^2(x+1) $$ Then I am ...
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2answers
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$G(B) = \{k \in \Bbb{Z} : \#(A+k) \cap B = \infty \text{ whenever } \#(A \cap B) = \infty\, A \subset \Bbb{Z}\}$ is a group. Question…

Let $B \subset \Bbb{Z}$ be any infinite subset. Define $G(B) = \{k \in \Bbb{Z} : \#(A+k) \cap B = \infty \text{ whenever } \#(A \cap B) = \infty\, A \subset \Bbb{Z}\}$. It forms an additive subgroup ...
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1answer
53 views

Show that the prime ideals above a prime $p$ are principal

Let $K=\mathbb Q(\alpha)$ where $ \alpha^3 -5\alpha + 5 = 0 $. I need to show that the prime ideals above 5 are principal, and find a generator for them. I have worked out the prime decomposition of ...
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3answers
30 views

$R$ is isomorphic to product of ideals $(a)\times (1-a)$

Following a question of mine on here, the answer says that if $a\in R$ (a commutative ring) is idempotent ($a^2=a$), then: $$R\simeq (a)\times (1-a)$$ I am trying to make sense of this by proving it. ...