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Questions tagged [ideals]

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, ideals in Lie algebras.

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Quotient ring of Gaussian integers

A very basic ring theory question, which I am not able to solve. How does one show that $\mathbb{Z}[i]/(3-i) \cong \mathbb{Z}/10\mathbb{Z}$. Extending the result: $\mathbb{Z}[i]/(a-ib) \cong \mathbb{...
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Comaximal ideals in a commutative ring

Let $R$ be a commutative ring and $I_1, \dots, I_n$ pairwise comaximal ideals in $R$, i.e., $I_i + I_j = R$ for $i \neq j$. Why are the ideals $I_1^{n_1}, ... , I_r^{n_r}$ (for any $n_1,...,n_r \in\...
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Why are maximal ideals prime?

Could anyone explain to me why maximal ideals are prime? I'm approaching it like this, let $R$ be a commutative ring with $1$ and let $A$ be a maximal ideal. Let $a,b\in R:ab\in A$. I'm trying to ...
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Classification of prime ideals of $\mathbb{Z}[X]$

Let $\mathbb{Z}[X]$ be the ring of polynomials in one variable over $\Bbb Z$. My question: Is every prime ideal of $\mathbb{Z}[X]$ one of following types? If yes, how would you prove this? ...
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Show that $\langle 2,x \rangle$ is not a principal ideal in $\mathbb Z [x]$

Hi I don't know how to show that $\langle 2,x \rangle$ is not principal and the definition of a principal ideal is unclear to me. I need help on this, please. The ring that I am talking about is $\...
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What are the left and right ideals of matrix ring? How about the two sided ideals?

What are the left and right ideals of matrix ring? How about the two sided ideals?
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Is $\mathbb{Z}[x]$ a principal ideal domain?

Is $ \mathbb{Z}[x] $ a principal ideal domain? Since the standard definition of principal ideal domain is quite difficult to use. Could you give me some equivalent conditions on whether a ring is a ...
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If $\mathop{\mathrm{Spec}}A$ is not connected then there is a nontrivial idempotent

I'm solving a problem from Atiyah-Macdonald. I have to show that if $X=\mathop{\mathrm{Spec}}A$ is not connected then $A$ contains idempotents $e \neq 0,1$. The converse is easy. If $e \in A$ is ...
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Maximal ideal in the ring of continuous functions from $\mathbb{R} \to \mathbb{R}$

Well, the problem I'm trying to solve is this: Let $A$ be the ring of all continuous functions from $\mathbb{R} \to \mathbb{R}$. Show that $M = \{f \in A: f(0)=0\}$ is a maximal ideal of $A$. I ...
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Structure of ideals in the product of two rings

$R$ and $S$ are two rings. Let $J$ be an ideal in $R\times S$. Then there are $I_{1}$, ideal of $R$, and $I_{2}$, ideal of $S$ such that $J=I_{1}\times I_{2}$. For me is obvious why $\left\{ r\in R\...
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Is quotient of a ring by a power of a maximal ideal local?

Say I have a commutative ring $R$ with a maximal ideal $m$. Then $m/m^k$ is a maximal ideal in $R/m^k$ for any $k$. Is it the only maximal ideal, i.e. is $R/m^k$ a local ring? This is a well known ...
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Finitely many prime ideals lying over $\mathfrak{p}$

Let $A$ be a commutative ring with identity and $B$ a finitely generated $A$-algebra that is integral over $A$. If $\mathfrak{p}$ is a prime ideal of $A$, there are only finitely many prime ideals $P$...
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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 is ...
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The ideal $I= \langle x,y \rangle\subset k[x,y]$ is not principal [closed]

The ideal $I= \langle x,y \rangle\subset k[x,y]$ is not a principal ideal. I don't know how to consider it. Any suggestions?
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Maximal ideals in $K[X_1,\dots,X_n]$

Let $K$ be a field, and $a_1,\dots,a_n \in K$. Prove that the ideal $$(X_1-a_1,\dots,X_n-a_n)$$ is maximal in $K[X_1,\dots,X_n]$. I tried proving that the only elements outside the ideal are the ...
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What do prime ideals in $k[x,y]$ look like?

Suppose that $k$ is an algebraically closed field. Then what do the prime ideals in the polynomial ring $k[x,y]$ look like? As far as I know, the maximal ideals of $k[x,y]$ are of the form $(x-a,y-b)$...
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An integral domain whose every prime ideal is principal is a PID

Does anyone has a simple proof of the following fact: An integral domain whose every prime ideal is principal is a principal ideal domain (PID).
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3answers
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Maximal ideals in the ring of real functions on $[0,1]$ [closed]

Let $S$ be the ring of all continuous functions from $[0,1]$ to $\mathbb R$. How to prove that all maximal ideals of $S$ have the form $M_{x_0}=\{f\in S \mid f(x_0)=0\}$? Thanks in advance.
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Why are ideals more important than subrings?

I have read that subgroups, subrings, submodules, etc. are substructures. But if you look at the definition of the Noetherian rings and Noetherian modules, Noetherian rings are defined with ideals ...
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In an extension of finitely generated $k$-algebras the contraction of a maximal ideal is also maximal

Let $k$ be a field and let $A \subset B$ be two finitely generated $k$-algebras. Prove that the contraction of any maximal ideal of $B$ is a maximal ideal of $A$. thank you very much again!
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An ideal that is maximal among non-finitely generated ideals is prime.

I've been doing some old exam problems and I've come across a problem that I've answered, but my gut is telling me that there's something I'm glossing over. Let $R$ be a commutative ring with ...
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An integral domain $A$ is exactly the intersection of the localisations of $A$ at each maximal ideal

This result appears to be ubiquitous as an algebra exercise. How do you prove this result? Let $A$ be an integral domain with field of fractions $K$, and let $A_{\mathfrak{m}}$ denote the ...
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Can an ideal in a commutative integral domain be its own square?

If $I$ is a non-zero proper ideal of a commutative integral domain, is it possible for $I$ to be its own square?
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1answer
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The set of all nilpotent elements is an ideal

Given that R is commutative ring with unity, I want show that set of all nilpotent elements is an ideal of R. I know how to show ideal if set is given but here set is not given to me. Can anyone ...
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1answer
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Ideals in direct product of rings [duplicate]

I am trying to solve this problem: Let $ R_1,...,R_n$ be rings with identity. Every ideal of $R=\prod_{i=1}^n R_i$ is of the form $\prod_{i=1}^n I_i$ where $ I_i$ is an ideal of $R_i$. The ...
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1answer
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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 have ...
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3answers
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Ideals of $\mathbb{Z}[X]$

Is it possible to classify all ideals of $\mathbb{Z}[X]$? By this I mean a preferably short enumerable list which contains every ideal exactly once, preferably specified by generators. The prime ...
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Explaining the product of two ideals

My textbook says that the product of two ideals $I$ and $J$ is the set of all finite sums of elements of the form $ab$ with $a \in I$ and $b \in J$. What does this mean exactly? Can you give examples?
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Height and minimal number of generators of an ideal

How can I determine the height and the least number of generators of the ideal $ I=(xz-y^2,x^3-yz,z^2-x^2y) \subset K[x,y,z] $? I tried to calculate the dimension of the vector space $I/I\mathfrak ...
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In a finite ring extension there are only finitely many prime ideals lying over a given prime ideal [duplicate]

I'm trying to solve the exercise 6.7 of Miles Reid's Undergraduate Commutative Algebra (pag 93). How can I prove that if $B$ is a finite ring extension of $A$, there are only finitely many prime ...
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Intuition behind “ideal”

To briefly put forward my question, can anyone beautifully explain me in your own view, what was the main intuition behind inventing the ideal of a ring? I want a clarified explanations in these ...
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Complement of maximal multiplicative set is a prime ideal

Let $R$ be a commutative ring with identity. I've been trying to prove the following: If $S \subset R$ is a maximal multiplicative set, then $R \setminus S$ is a prime ideal of $R$. I have spent ...
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1answer
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What's the motivation of the definition of primary ideals?

$$xy\in\mathfrak q\:\Rightarrow\:\text{either $x\in\mathfrak q$ or $y^n\in\mathfrak q$ for some $n\gt0$}.$$ Primary ideals can be regard as the generalization of prime ideals and radical. But why it'...
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Construct ideals in $\mathbb Z[x]$ with a given least number of generators

How do you construct, for each $n\geq 1$, an ideal in $\mathbb Z[x]$ of the form $(a_1,a_2,\dots,a_n)$ with $a_i\in \mathbb Z[x]$ such that it is impossible to have $(b_1,b_2,\dots,b_m)=(a_1,a_2,\dots,...
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Why is $(2, 1+\sqrt{-5})$ not principal?

Why is $(2, 1+\sqrt{-5})$ not principal in $\mathbb{Z}[\sqrt{-5}]$? Say $(2,1+\sqrt{-5})=(\alpha)$, then since $2\in(2,1+\sqrt{-5})$ we have $2\in (\alpha)$, so $\alpha\mid2$ in $\mathbb Z[\sqrt{-5}]$...
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1answer
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if R is a commutative ring in which all the prime ideals are finitely generated then R is Noetherian [duplicate]

Prove that if $R$ is a commutative ring in which all the prime ideals are finitely generated, then $R$ is Noetherian. Here is what I been told to do: Suppose that $R$ is not Noetherian, and use Zorn’...
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Can the square of a proper ideal be equal to the ideal?

Let $R$ be a ring, commutative with $1$, let $\mathfrak{i}$ be an ideal, not the whole ring. In general $\mathfrak{i}^2\subseteq\mathfrak{i}$. Can this inclusion be an equality, or it is always a ...
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1answer
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Cardinality of quotient ring $\mathbb{Z_6}[X]/(2X+4)$

Let $R$ be the ring obtained by taking the quotient of $\mathbb{Z_6}[X]$ by principal ideal $(2X+4)$. Then 1) $R$ has infinite elements 2) $R$ is field 3) $5$ is unit in $R$ 4) $4$ is unit in $R$. ...
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Prime ideals in $C[0,1]$

Are there any prime ideals in the ring $C[0,1]$ of continuous functions $[0,1]\rightarrow \mathbb{R}$, which are not maximal? Perhaps, I duplicate smb's question, but this is an interesting problem! ...
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Converse to Chinese Remainder Theorem

So as seen on this question Converse of the Chinese Remainder Theorem, we know that if $(n,m) \neq 1$, then $\mathbb{Z} /mn \mathbb{Z} \ncong \mathbb{Z}/n\mathbb{Z} \times \mathbb{Z}/m\mathbb{Z}$, ...
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Unique factorization domain and principal ideals

If R was a unique factorization domain, can we deduce that for a nonzero element d in R, d has a finite number of divisors? I need this in solving this question "If R is a unique factorization ...
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In a principal ideal domain, prove that every non trivial prime ideal is a maximal ideal. What could be wrong in this approach?

In a principal ideal domain, prove that every non trivial prime ideal is a maximal ideal Attempt: Let $R$ be the principal ideal domain. A principal ideal domain $R$ is an integral domain in which ...
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Number of elements in the ring $\mathbb Z [i]/\langle 2+2i\rangle$

The question is : Show that $I=\langle 2+2 i\rangle$ is not a prime ideal of $\mathbb Z[i]$. Also find the number of elements in $\mathbb Z[i]/I$ and its characteristic. My try: I started with ...
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Finitely generated idempotent ideals are principal: proof without using Nakayama's lemma

I am trying to understand Nakayama's lemma. It looks like some "fixed point theorem". Using Nakayama's lemma , I can easily solve the following question. I want another proof. Thanks. Let $A$ be a ...
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1answer
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Ring of formal power series over a field is a principal ideal domain

If $K$ is a field, any non-zero ideal in the ring of formal power series $A=K[[X]]$ is of the form $AX^n$ with $n\geq 0$, so $A=K[[X]]$ is a principal ideal ring. I can't see why. Usually when we ...
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Prove that the preimage of a prime ideal is also prime.

Let $f: R \rightarrow S$ be a ring homomorphism, with $R$ and $S$ commutative. If $P$ is a prime ideal of $S$, show that the preimage $f^{-1}(P)$ is a prime ideal of $R$. Define $g: S \rightarrow S/...
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3answers
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Cardinality of the quotient ring $\mathbb{Z}[x]/(x^2-3,2x+4)$ [duplicate]

This problem is from a practice exam I was working on. What is the cardinality of the quotient $\mathbb{Z}[x]/(x^2-3,2x+4)$ ? Thoughts. If I find a ring that is easier to handle then this then I ...
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3answers
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An ideal whose radical is maximal is primary

I've got to prove that an ideal $Q$ whose radical is a maximal ideal is a primary ideal. That is, I want to prove that if $xy\in Q$, then $x\in Q$ or $y^n\in Q$ for some $n>0$. I've been trying ...
5
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1answer
394 views

Primary ideals in Noetherian rings

For an $R$-module $M$ I have the following definition for a submodule $N\subset M$ to be $\mathfrak{p}$-primary: this is the case when $\text{Ass}(M/N) = \{\mathfrak{p}\}$, that is, $M/N$ is coprimary ...
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1answer
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Showing $\{a+b\sqrt{2} \in R$ | $a$ is divisible by $2\}$ is an ideal.

This was a problem in my math book and I was wondering what the proof looks like: Let $R = \{a + b\sqrt{2} | a,b\in \mathbb{Z}\}$. Show that $I = \{a+b\sqrt{2}\in R | a$ is divisible by $2\}$ is an ...