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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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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1answer
57 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, ...
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1answer
62 views

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 ...
4
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1answer
45 views

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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1answer
31 views

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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0answers
37 views

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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3answers
29 views

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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4answers
41 views

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} ...
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1answer
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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0answers
37 views

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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2answers
59 views

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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1answer
41 views

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 $$ I(L_{1}\cup L_{2})=(X_{0}X_{2},X_{0}X_{3},X_{1}X_{2},X_{1}X_{3}). $$ ...
2
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1answer
217 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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2answers
1k views

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 ...
2
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1answer
46 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 ...
0
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1answer
40 views

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> ...
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3answers
128 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 ...
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1answer
61 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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0answers
41 views

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 $$\phi(e_i) = m_i.$$ ...
0
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0answers
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 ...
2
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2answers
30 views

$G(B) = \{k \in \Bbb{Z} : \#(A+k) \cap B = \infty \text{ whenever } \#(A \cap B) = \infty\, A \subset \Bbb{Z}\}$ 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 ...
3
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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. ...
0
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2answers
29 views

Finding all ideals N of $Z_{12}$ and compute $Z_{12}/N$ in each case.

I already find all the ideals N of $Z_{12}$. Here's what I've got: Since ideals must be additive subgroups, by group theory we see that the possibilities are restricted to the cyclic additive ...
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3answers
52 views

Cardinality of the quotient of $M_2(\mathbb{Z})$ by the ideal of matrices with even entries

Let $R=\left\{\begin{pmatrix} a_1&a_2\\a_3&a_4 \end{pmatrix} \mid a_i\in \mathbb{Z} \right\}$ and let $I$ be the subset of $R$ with even entries. Show that $I$ is an ideal of $R$. What is ...
2
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1answer
25 views

Finding ideals of $F_2[C_2]$

I'm trying to find the ideals of $F_2[C_2]$ I believe the elements are $(0,1,x) $ So far I have the ideal {0} I can't seem to spot any others, have I made a mistake or missed something?
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28 views

Are there any uses of this notion of ideal? $xy + k \in (p) \implies x + k$ or $y + k \in (p)$

Abstracting the definition of prime ideal a little we have ideals $(p) \subset \Bbb{Z}$ such that $xy + k \in (p) \implies x + k \in (p)$ or $y + k \in (p)$. For example, taking $k = 1$, and $p = 2$ ...
0
votes
1answer
62 views

Show $(3,x)$ is a principal ideal in $\mathbb{Z}_{6}[x]$.

Since $R[x]/(I)=(R/I)[x]$, and by the Chinese Remainder Theorem, $$\mathbb{Z}_6[x]/(3,x) \cong \mathbb{Z}_3[x]/(3,x) \times \mathbb{Z}_2[x]/(3,x) \cong ...
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0answers
16 views

intersection of principal ideals in commutative rings with unity [duplicate]

Consider a commutative ring with unity. The intersection of two principal ideals is an ideal but not necessarily a principal ideal. (http://commalg.subwiki.org/wiki/Principal_ideal) However in ...
0
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2answers
59 views

Proving that particular ideal is the kernel of a homomorphism of polynomial rings

I have the homomorphism $f:\mathbb{R}[x,y,z]\rightarrow\mathbb{R}[t]$ with $f(x)=t,f(y)=t^2,f(z)=1$, and I want to prove that $\ker f=(x^2-y,z-1)$. It is obvious that this ideal is in kernel, but ...
2
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1answer
56 views

Compute the decomposition of $5\mathbb{Z}_K$ as a product of prime ideals

Let $K = \mathbb{Q}(\alpha)$ such that $\alpha^3 - 5\alpha + 5 = 0$. It is easy to show that $\mathbb{Z}_K = \mathbb{Z}[\alpha]$ and that $5$ is not maximal in $\mathbb{Z}[\alpha]$. So we cannot use ...
2
votes
1answer
61 views

$R$ be a commutative ring with unity such that every prime ideal contains no non-zero zero divisor , then is $R$ an integral domain?

Let $R$ be a commutative ring with unity such that every prime ideal contains no non-zero zero divisor (i.e. if $P$ is a prime ideal and $x,y \in P$ with $xy=0$ then either $x=0$ , or $y=0$). Then ...
0
votes
1answer
28 views

Compact operators form the only closed proper ideal of bounded linear operators

I am trying to understand the following proof in Trace Ideals and Their Applications by Barry Simon (Proposition 2.1): Let $\mathcal{J}$ be a two-sided ideal in $\mathcal{L}(\mathcal{H})$ containing ...
0
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2answers
67 views

Ideal $I=({ X }^{ 2 },2X)$ of $\mathbb Z[X]$ generated by ${ X }^{ 2}$ and $2X$ is not primary.

Let $I=({ X }^{ 2 },2X)$ ideal of $\mathbb Z[X]$ generated by ${ X }^{ 2}$ and $2X$. Show that $I$ is not primary. I tried to find $$\sqrt { I } =\sqrt { ({ X }^{ 2 })+(2X) } =\sqrt { \sqrt { { ...
2
votes
3answers
87 views

Proving that there is no ideal $I$ in $\mathbb{Z}[t]$ such that $\mathbb{Z}[t]/I \cong \mathbb{Q}$ [duplicate]

I want to prove that there is no ideal $I$ in $\mathbb{Z}[t]$ such that $\mathbb{Z}[t]/I \cong \mathbb{Q}.$ The first part of the question asks us to show that if $\phi$ is a nonzero ring ...
1
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2answers
41 views

Identities of the spectrum of a ring

Let $R$ be a commutative ring, then the spectrum of $R$ is a topological space: $$Spec(R)=\{ P:P \;\text{is a prime ideal of R} \}$$ where the close sets are $$V_{I}=\{P \in Spec(R) \;\text{such ...
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0answers
14 views

Example(s) of tertiary ideals?

Could anyone provide me with some specific example of a tertiary ideal, illustrating why it is so? Thanks in advance.
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1answer
30 views

Is it possible to know the prime factors of $3+5i$?

Let $\mathbb{Z}[i] = \{a+bi : a,b \in \mathbb{Z}\}$ and $\mathbb{Q}(i) = \{a+bi : a,b \in \mathbb{Q}\}$. Find $\alpha \in \mathbb{Z}[i]$ such that $(3+5i,1+3i) = (\alpha)$ Since ...
1
vote
2answers
71 views

Is a function in an ideal? Verification by hand and Macaulay 2

Suppose $$f_1=-4x^4y^2z^2+y^6+3z^5,$$ $$f_2=-4x^2y^2z^2+y^6+3z^5,$$ $$f_3=4x^4y^2z^2+y^6+3z^5,$$ $$f_4=4x^2y^2z^2+y^6+3z^5$$ and $$I=\langle xz-y^2,x^3-z^2\rangle\subset\mathbb C[x,y,z].$$ Is ...
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votes
2answers
84 views

Prove that two ideals are equal [closed]

Prove that the ideals $I_1=\langle x+xy,y+xy,x^2,y^2\rangle$ and $I_2=\langle x, y\rangle$ are equal. I understand the concept of trying to show that $I_1 \subset I_2$ and vice versa, but I have ...
3
votes
3answers
46 views

$R$ be an infinite commutative ring with unity such that for every non-zero ideal $I$ , $R/I$ is finite ; then is $R$ a PID or at least Noetherian?

Let $R$ be an infinite commutative ring with unity such that for every non-zero ideal $I$ of $R$ , $R/I$ is finite; then is $R$ a PID or at least Noetherian ? I can only prove that $R$ must be an ...
1
vote
1answer
57 views

Extension of ideals in integral extensions

Let $R\subset S$ is an integral extension in the category of commutative rings with unity. I have three questions: 1) Is every ideal of $S$ an extended ideal? 2) Is extension of each idempotent ideal ...
0
votes
2answers
31 views

A commutative ring with identity is a field if and only it has no nonzero proper ideals [duplicate]

Obviously, if $F$ is a field, and $I$ is it's nonzero ideal, then it contains an invertible element of $F$(any nonzero element of $F$). Denote this element as $a$. Since $I$ is ideal, $aa^{-1} = 1 \in ...
6
votes
5answers
537 views

Example of a ring with an infinite inclusion chain of ideals [closed]

I'm trying to track down an example of a ring in which there exists an infinite chain of ideals under inclusion. (i.e. $I_1 \subsetneq I_2 \subsetneq I_3 \subsetneq...$)
0
votes
0answers
40 views

Let $I$ be an ideal of a ring $R$. Then Show that $|(R/I)| = 1 $ if and only if $R = I$.

(i) Show that $|(R/I)| = 1 $ if and only if $R = I$. (ii) Show that if $R$ has an identity 1 then (if $I \neq R$) so does $R/I$, and if $R$ is commutative, then so is $R/I$. I know that the quotient ...
4
votes
1answer
210 views

Is the irreducibility of a ring preserved by localization at a prime ideal?

Let $R$ be a commutative ring and $\mathfrak p $ a prime ideal of $R$. Suppose that $R$ satisfies the following property: the intersection of two nonzero ideals is always nonzero. Is the property also ...
1
vote
1answer
38 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 ...
1
vote
1answer
39 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 ...
1
vote
1answer
27 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 ...
0
votes
1answer
44 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 ...