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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Ideals, closure

Suppose $x\in R$, where $R$ is a commutative ring with unity, and $I$ is an ideal of $R$. Suppose further that $rx\in I$ for every $r\in R$. Does this imply that $x \in I$? (if so why?) Thanks for any ...
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Union of Chain of Ideals

I'm writing a project in a "Rings and Modules" course, and I've come across the following proposition, stated without proof: Proposition 1.2. In a commutative ring R , the product of ideals is ...
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59 views

Image of homomorphism from ideal is ideal [duplicate]

Let $A,B$ be rings. If $f:A\rightarrow B$ is a homomorphism from $A$ onto $B$ with kernel $K$, and $J$ is an ideal of $A$ such that $K\subseteq J$, then $f(J)$ is an ideal of $B$: My solution: Let ...
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28 views

family of ideals

Can somebody explain what exactly is defined to be family of ideals. Is it just an arbitrary collection of ideals of a ring or is there some structure is this family? Thank you
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Ideal iff closed with respect to addition and absorbs product

If $A$ is a ring (with unity), prove that $J$ is an ideal of $A$ if and only if $J$ is closed with respect to addition and $J$ absorbs products in $A$. If $J$ is an ideal of $A$, then by ...
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An element does not belong to an ideal

How can I prove that the element $x-5$ does not belong to the ideal $(x^2-25,-4x+20)$ in $\mathbb Z[x]$. I tried to show that by proving $x-5\neq(x^2-25)f(x)+(-4x+20)g(x)$ for all $f,g$. Any ...
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prime ideal that is not max ideal

It's easy to see that $\mathbb{Z}$ is noetherian but not artinian. In my course notes there's proven: $A$ a commutative ring: $A$ artininian $\Leftrightarrow$ $A$ noetherian and Spec($A$) = Max($A$) ...
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Graded Betti Numbers of a Graded Ideal with Linear Quotients

Exercise 8.8(a) in Monomial Ideals by Herzog and Hibi: Let $I\subset S=K[x_{1},...,x_{n}]$ be a graded ideal which has linear quotients with respect to a homogeneous system of generators ...
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Graded Betti Numbers of a Stable Monomial Ideal

Exercise 8.8 in Monomial Ideals by Herzog and Hibi: Let $I\subset S=K[x_{1},...,x_{n}]$ be a stable monomial ideal with $G(I)=\{u_{1},...,u_{m}\}$ and such that for $i<j$, either ...
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P-primary Monomial Ideal

Let $P=(x_{1},...,x_{r})\subset S=K[x_{1},...,x_{n}].$ Show that a monomial ideal $Q$ is $P$-primary if and only if there exists a monomial ideal $Q'\subset T=K[x_{1},...,x_{r}]$ such that ...
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Exercise 1.2.4 and Example 4.3.6 in Liu

I want to prove that if $X$ is a noetherian scheme then any flat closed immersion into $X$ is open, that is, if $A$ is noetherian then $\varphi:\operatorname{Spec}(A/I)\to\operatorname{Spec}(A)$ is ...
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Special basis of an ideal as a $\mathbb{Z}$-module in number fields

I was speculating that the following may be true (but do not see any easy way to settle it; it must be known, I suppose): Given a (say, prime) ideal $\mathfrak{p}$ of the ring of integers ...
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3answers
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Question regarding ideals and vector spaces

this is my first time I am posting on this forum. My question is regarding a sentence I read on page 27 of "Algebraic Number Fields" by "Gerald J. Jansuz". The set-up is as follows: Let $R \subset ...
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coprime elements

Let $R$ be a ring, then two elements $I,J$ are coprime, if $RJ+RI=R$ or in other words, if there exist $r_1,r_2 \in R$ such that $r_1I+r_2J=u$, where $u$ is a unitity in $R$. Now let $\mathbb{Q}$ be ...
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77 views

An equivalence for $\operatorname{grade}(I,M)\ge 2$.

Let $R$ be a Noetherian ring, $M$ a finite $R$-module and $I$ an ideal of $R$. Show that $\operatorname{grade}(I,M)\ge 2$ iff the natural homomorphism $M \mapsto\operatorname{Hom}_R(I,M)$ is an ...
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Prove the intersection of localizations at maximal ideals is $A$

Let $A$ be an integral domain with field of fractions $K$. How can I prove that $\bigcap\{A_{m}\mid m \subseteq ​​A\text{ is maximal}\} = A$?
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Why $I=\left\{p(x)\in \mathbb{Z}\left[X\right]:2\mid p(0)\right\}$ is not a principal ideal? [duplicate]

I saw this question but I still do not understand: What is the difference between ideal and principal ideal? At my homework I had to prove to things about $I=\left\{p(x)\in ...
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46 views

A property of $I$-adic topologies

Let $R$ be a commutative ring with multiplicative identity and $I$ a proper ideal of $R$ such that the intersection of its powers is the zero ideal. It can be shown that if the $I$-adic topology is ...
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$(R_1\oplus R_2) [x]/(p(x)) = R_1[x]/(p(x))\oplus R_2[x]/(p(x))$?

For convenience, I shall use '$=$' to denote isomorphisms. Suppose we have a commutative ring $R = R_1\oplus R_2$, and $(p(x))$ is the ideal generated by $p(x)\in R[x]$. Can we deduce that ...
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Irreducible homogeneous ideals

I have the following question: Let $I$ be a homogeneous ideal. Is it true that $I$ is irreducible if and only if it can't be written as the intersection of two homogeneous ideals? So, is it ...
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1answer
108 views

Primary decomposition of a monomial ideal

Can anyone give me an idea about the primary decomposition of the ideal $I=(x^3y,xy^4)$ of the ring $R=k[x,y]$? I am trying to connect the primary decomposition with the set Ass(R/I) which i ...
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Is an ideal which is maximal with respect to the property that it consists of zero divisors necessarily prime?

This is in follow-up to this question. Let $R$ be a commutative ring with identity and consider the set $Z \subset R$ of zero divisors. If the ideal $I\subset Z$ is maximal with respect to the ...
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Ideal Generated by $H\cup K$ [duplicate]

If $H$ and $K$ are ideals of a ring $R$, what are the elements of $\langle H\cup K\rangle$? What I am trying to ask is that how does its element look?
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to prove that the set consisting of all zero divisors in a commutative ring with unity contains at least one prime ideal

im asked to prove that the set consisting of all zero divisors in a commutative ring with unity contains at least one prime ideal. i cant even start in the proof , ive just defined my set but cant ...
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the direct image of an ideal needs not to be an ideal

I need an example of a ring mapping homo such that the image of an ideal needs not to be an ideal ? I found that the image is an ideal if the mapping was an onto one ! so all we need to find a mapping ...
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Prove or disprove that the ideal $(2+4\mathbb{Z},x)$ is a principal ideal in $(\mathbb{Z}/4\mathbb{Z})[x]$

Prove or disprove that the ideal $(2+4\mathbb{Z},x)$ is a principal ideal in $(\mathbb{Z}/4\mathbb{Z})[x]$. I know $\mathbb{Z}/4\mathbb{Z}$ is not a field. have something help?
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Saturation and Associated Primes of an Ideal

If $I$ is an ideal of a Noetherian ring $S$ and $x,y\in S$, show that the following are equivalent: $(1)$ $(I:y^{\infty})=(I:(x,y)^{\infty})$ $(2)$ Every associated prime of $I$ that ...
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77 views

Suppose that $I$ is an ideal of $R$ which is maximal with respect to the property that it is proper and not prime.

Let $R$ be a commutative ring with $1$. Suppose that $I$ is an ideal of $R$ which is maximal with respect to the property that it is proper and not prime. Deduce that $I$ is contained in at most ...
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About the Linear Quotients of Square of an Ideal with Linear Quotients

Let $I$ be a monomial ideal generated by quadratic monomials $u_{1},...,u_{s}$ and suppose that $I$ has linear quotients with respect to this given ordering. Is it true or false that $I^{2}$ has ...
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88 views

Ideal Generated by the Union of Two Ideals

Let $I$ and $J$ be ideals of a ring $R$. Prove that $I+J$ is an ideal of $R$ and that $I+J=\langle I\cup J\rangle$, the ideal of $R$ generated by $I\cup J$.
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Ideals containing $(6, x^3-1)$ in $\mathbb{Z}[x]$

I want to find all ideals containing $(6, x^3-1)$ in $\mathbb{Z}[x]$ and I can only find ten ideals: $\mathbb{Z}[x]$ $(2, x-1),\; (2, x^2+x+1),\; (3, x-1)$ $(6,x-1),\; (2,x^3-1),\; (6, x^2+1+1),\; ...
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Let $R$ be a commutative ring with $1$. Suppose that every nonzero proper ideal of $R$ is maximal. Prove that there are at most two such ideals.

Let $R$ be a commutative ring with $1$. Suppose that every nonzero proper ideal of $R$ is maximal. Prove that there are at most two such ideals. Help me some hints. I have no idea to start. ...
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42 views

Kernel of $f$ is a maximal ideal in $R$

Let $R = \{a + b \sqrt{-3} \mid a,b \in \mathbb{Z}\}$ and define $f: R \to \mathbb{Z}/7\mathbb{Z}$ as $f(a+b\sqrt{-3}) = a - 2b + 7 \mathbb{Z}$. Show that $f$ is surjective and that the kernel of $f$ ...
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A commutative ring whose all proper ideals are prime is a field. [closed]

Let $R$ be a commutative ring with $1$. Suppose that all ideals $I \neq R$ are prime. Prove that $R$ is a field. Help me some hints. Thanks a lot.
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Isomorphism of K-algebras

Let $k[x]$ be a polynomial ring and $I$ an ideal (resp. graded). If $k\subset K$ is a field extension, then prove that there is a natural (resp. graded) isomorphism of $K$-algebras: $$K[x]/IK[x]\cong ...
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1answer
87 views

Saturation of a Graded Radical Ideal in $S=k[x_{1},…,x_{n}]$

Let $I\subset S=k[x_{1},...,x_{n}]$ be a graded radical ideal different from $\mathfrak{m}=(x_{1},...,x_{n})$. Prove that $I$ is saturated. To prove that $I$ is saturated it is sufficient to ...
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1answer
67 views

Is $\Pi_{I} \mathbb{Z}$ a principal ideal ring?

Why is $\Pi_{i=1}^{\infty} \mathbb{Z}$ a PID principal ideal ring? (Edit: the OP later revealed that they had not meant to include "domain." Edited to reflect that.) Actually, I saw this statement ...
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1answer
99 views

Example of non-unique factorization domain that satisfies ACC on principal ideals

We know the theorem that an integral domain $R$ is UFD if and only if (1) $R$ satisfies the ascending chain condition for principal ideals; (2) every irreducible element is prime. Now I ...
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Multiple choice question regarding the ideal of $C[0, 1]$

Let $C[0,\ 1]$ be the ring of continuous real-valued functions on $[0,\ 1]$, with addition and multiplication defined pointwise. For any subset $S$ of $C[0,\ 1]$ let $$Z(S) =\{f \in C[0,\ 1] \mid f(x) ...
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When does $IJ=(I+J)(I\cap J)$

It's clear that $IJ\supset(I+J)(I\cap J)$, but when is the reverse inclusion true? So far, the simplest counterexample I could find was $I=(x), J=(x^2,y)$ in $k[x,y]$.
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Verifying that the ideal $(x^3-y^2)$ is prime

How to prove that the ideal $I=(x^3-y^2)$ in $k[x,y]$ is prime? I have constructed a map from $k[x,y]$ to $k[t]$, which maps $x$ to $t^2$, and $y$ to $t^3$. Then, I want to show that the kernel ...
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Give an infinite sequence of principal ideals of $R$ such that the ascending chain condition does not hold

Let $R=\{\sum_{i=0}^n a_ix^i\mid n\geq 0, a_0\in\mathbb{Z}, a_i\in\mathbb{Q} \text{ for } i\geq 1\}$. Give an infinite sequence of principal ideals of $R$ such that the ascending chain condition ...
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Minimal Ideal of a Commutative Ring with Unity

Can anyone help me prove this? This one is from Malik's Fundamentals of Abstract Algebra. An ideal $I$ of a ring $R$ is called a minimal ideal if $I≠{0}$ and there does not exist any ideal $J$ of R ...
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48 views

Qustion about Ideal…(Ring theory)

I know that if $I,J$ are Ideals of $R$ so $I+J=\left\{i+j|i\in I, j\in J\right\}$ is Ideal to... I need to find $a\in \mathbb{Z}$ s.t. ...
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Unital commutative ring and distinct maximal ideals.

Let $R$ be a unital commutative ring and $m_1,m_2$ distinct maximal ideals. Prove that $$\frac{R}{m_1m_2}\simeq\frac{R}{m_1} \times \frac{R}{m_2}.$$ I think something like this homomorphism ...
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127 views

Are the two infima equal?

Let $R$ be a ring (not necessarily commutative) or an algebra over some field with a norm defined on it and let $I$ be an ideal in $R$. Let $a,b \in R$. Does it hold that $\inf_{i,j \in I}\|ab + ai ...
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Product of two maximal ideals in a commutative ring with unity is equal to their intersection. [closed]

Let $R$ be a commutative ring with unity and let $m_1$ and $m_2$ be two different maximal ideals in $R$. Prove: $m_1 m_2 = m_1 \cap m_2$. Find an example of two different prime ideals $P_1$ and $P_2$ ...
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58 views

Sanity check about basic abstract algebra

In the following I am talking about algebras over $\mathbb C$ and ideals. An ideal is called modular if there is $u \in A$ so that for all $a \in A$ both $a-ua$ and $a-au$ are in the ideal $I$. I read ...
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2answers
64 views

$\operatorname{Ann}(\operatorname{Ann}(N))=N$ for a submodule $N$ of an $R$-module?

Suppose $R$ is a commutative and unitary ring and $M$ is an $R$-module. Is it generally true that for any submodule $N$ in $M$, $\operatorname{Ann}(\operatorname{Ann}(N))=N$? If it's not true in ...
6
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1answer
79 views

P(R) is contained in Nil(R) for noncommutative rings.

How to show that $P(R)$ is contained in $\operatorname{Nil}(R)$ (where $R$ is a noncommutative ring with identity)? Definitions I am using: A nil right ideal is one whose elements are all ...