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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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 ...
3
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2answers
598 views

In a finite ring extension there are only finitely many prime ideals lying over a given prime ideal

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 ...
16
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3answers
550 views

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 $e \in A$ ...
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2answers
892 views

Methods to check if an ideal of a polynomial ring is prime or at least radical

I am looking for methods to check whether a given ideal in $K[x_0,\dots,x_n]$ is prime. I mean something you can effectively use in some concrete non-trivial example. To be more explicit, I am working ...
6
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1answer
489 views

Is each power of a prime ideal a primary ideal?

I want to show that each power of a prime ideal is a primary ideal or I have to think about a counterexample?
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6answers
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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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2answers
2k views

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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3answers
705 views

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 ...
11
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3answers
536 views

Examples of rings with ideal lattice isomorphic to $M_3$, $N_5$

In thinking about this recent question, I was reading about distributive lattices, and the Wikipedia article includes a very interesting characterization: A lattice is distributive if and only if ...
3
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1answer
287 views

Is the number of prime ideals of a zero-dimensional ring stable under base change?

Let $A$ be a zero-dimensional ring of finite type over a field $k$ and let $X= \textrm{Spec} \ A$ be its spectrum. Note that $X$ is a finite set. Suppose that $k\subset K$ is a finite field extension ...
11
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3answers
912 views

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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3answers
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$\mathbb Z\times\mathbb Z$ is principal but is not a PID

I need to find an example of a ring that is not a PID but every ideal is principal. I know that $\mathbb Z\times\mathbb Z$ is not an integral domain, so certainly is not a PID, but here every ideal is ...
4
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2answers
304 views

Cardinality of the quotient ring $\mathbb{Z}[x]/(x^2-3,2x+4)$

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 ...
2
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2answers
468 views

Maximal ideals in the ring of real functions on $[0,1]$

Assume $S$ to be 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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8answers
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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 ...
6
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4answers
786 views

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).
8
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5answers
824 views

If the localization of a ring $R$ at every prime ideal is an integral domain, must $R$ be an integral domain?

Let $R$ be a commutative ring. Suppose that for every prime ideal $p$ of $R$, the localized ring $R_p$ is an integral domain. Must $R$ be a integral domain? I was trying to think of counter-examples, ...
8
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2answers
461 views

Infinite product of fields

The main source of inspiration for this question is this excerpt Recall: An ultrafilter on the set X gives you a maximal ideal in the ring of all real-valued functions, and these are the only ...
4
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1answer
403 views

Is $(XY - 1)$ a maximal ideal in $k[[X]][Y]$?

Is $(XY - 1)$ a maximal ideal in $k[[X]][Y]$, and if so, how can I see it? It is at least prime because the generator is irreducible, and by the same argument it is maximal among all principal ...
3
votes
1answer
168 views

Norm of ideals in quadratic number fields

I do not really understand how to factor ideals in a quadratic field $K = \mathbb{Q}(\sqrt{d})$, mainly because I have some trouble computing the norm of ideals. I think I understand what is going on ...
3
votes
1answer
956 views

If $I$ is a maximal ideal of $R$, why is $R/I$ a field?

If $I$ is a maximal ideal of $R$, why is $R/I$ a field? I'm trying to use the fact that $I$ is maximal to show that $R/I$ only have ideals $\{0\}$ and $R/I$. Can anyone help me with this method. Many ...
2
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2answers
440 views

Prove that every nonzero prime ideal is maximal in $\mathbb{Z}[\sqrt{d}]$

$d \in \mathbb{Z}$ is a square-free integer ($d \ne 1$, and $d$ has no factors of the form $c^2$ except $c = \pm 1$), and let $R=\mathbb{Z}[\sqrt{d}]= \{ a+b\sqrt{d} \mid a,b \in \mathbb{Z} \}$. ...
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5answers
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The image of an ideal under a homomorphism may not be an ideal

This is an elementary question about ideals. Consider a ring homomorphism $$ f: \mathbb{Z} \rightarrow \mathbb{Z}[x], $$ and consider the ideal $\left< 2\right>$ in $\mathbb{Z}$. When why is ...
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1answer
175 views

Number of generators of the maximal ideals in polynomial rings over a field

Hi I'm trying to prove the following If $K$ is a field (not necessary algebraically closed) then every maximal ideal of $K[x_{1},\dots,x_{n}]$ is generated by exactly $n$ elements. I know that ...
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3answers
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A proof that this set is an ideal of a commutative ring

This is a homework problem which I have worked hard on, but got stuck at the last step. Any assistance would be much appreciated. The problem is from Herstein's Abstract Algebra, 3rd ed., section 4.3, ...
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2answers
140 views

Is $Z(R)$ a maximal ideal?

If $p$ and $q$ are two maximal ideals in the set of zero-divisors in a ring $R$ with non-zero intersection between $p$ and $q$. does the set of all zero-divisors are a maximal ideal and equal the ...
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1answer
461 views

Maximal ideals in $R[x]$

I want to prove the following result: Let $R$ be a ring and $M$ a maximal ideal in $R$. If $P$ is a prime ideal in $R[x]$ that (strictly) contains $M[x]$, then $P$ is a maximal ideal in $R[x]$. ...
4
votes
2answers
187 views

What is a projective ideal?

I've been looking for the definition of projective ideal but haven't found anything, all I've seen is the definition of projective module (but I don't know how these are related, if they are ¿?). Does ...
3
votes
1answer
218 views

Ideals in the ring of endomorphisms of a vector space of uncountably infinite dimension.

I know that if $V$ is a vector space over a field $k,$ then $\operatorname{End}(V)$ has no non-trivial ideals if $\dim V<\infty;$ $\operatorname{End}(V)$ has exactly one non-trivial ideal if ...
2
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1answer
414 views

Preimage of a maximal ideal.

My textbook says that if $f: R \rightarrow S$ is a ring homomorphism, where R and S are commutative; then if P is a maximal ideal of S, it might not necessarily be a maximal ideal of R. A ...
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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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1answer
60 views

Describe units and maximal ideals in these two PIDs

If $p$ is a fixed prime integer, let $R$ be the set of all rational numbers that can be written in the form $(a)$ $\frac{a}{b}$ with $b$ not divisible by $p$. $(b)$ $\frac{a}{b}$ with $b=p^k$ for a ...
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0answers
154 views

Maximal ideal in commutative ring

Let $R$ be a Dedekind domain, $\mathfrak{m}$ be a maximal ideal in $R[x]$ is of the form $\mathfrak{m} = (\mathfrak{p},f(x))$ where $\mathfrak{p}$ is a maximal ideal in $R$, and $f$ is a polynomial in ...
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4answers
323 views

Prime ideal and nilpotent elements

If $\mathfrak p \subset R$ is a prime ideal, prove that for every nilpotent $r \in R$ it follows that $r \in \mathfrak p$. The only hint that my tutor gave me was to use induction. Can someone ...
1
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1answer
269 views

Annihilator of a simple module

Let $R$ be a finitely generated commutative ring and $C$ an $R$-algebra ($C$ is not necessarily commutative). Assume that $C$ is a finitely generated $R$-module. If $S$ is a simple $C$-module, then ...
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1answer
92 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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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 ...
15
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0answers
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Class group and factorizations

There is a common characterization of the class group ${\rm Cl}(R)$ as a kind of measure of how badly factorization fails to be unique. The most obvious justification for this sentiment is that the ...
11
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3answers
310 views

What is the quotient $\mathbb Z[\sqrt{3}]/(1+2\sqrt{3})$?

I am currently doing a past paper and it asks the following: Prove that for $I=(1+2\sqrt{3})$ we have $\mathbb Z[\sqrt{3}]/I$ a field with $11$ elements. If I assume standard algebraic number ...
6
votes
4answers
158 views

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. ...
5
votes
2answers
929 views

When is a product of two ideals strictly included in their intersection?

Let $I,J$ two ideals in a ring $R$. The product of ideals $IJ$ is included in $I \cap J$. For example we have equality in $\mathbb{Z}$ if generators have no common nontrival factors, in a ring $R$ ...
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4answers
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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 $A$ be a maximal ideal. Let $a,b\in R:ab\in A$ I'm trying to ...
4
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3answers
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Ideal of the twisted cubic

The twisted cubic is the image of the morphism $\phi : \mathbb{P}^1 \to \mathbb{P}^3 , (x:y) \mapsto (x^3:x^2 y:x y^2:y^3)$, it is given by $X = V(ad-bc,b^2-ac,c^2-bd)$. Now I would like to compute ...
4
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1answer
159 views

Ascending chain conditions on homogeneous ideals

Here is one exercise from some notes on graded rings. I tried but I got no idea to solve it. Please help me. Thanks. Let $R$ be a graded ring. Prove that $R$ is Noetherian (Artinian) if and only ...
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2answers
356 views

On proving every ideal of $\mathbb{Z}_n$ is principal

I was working on a problem in Robert Ash's Abstract Algebra, and didn't follow part of the solution. The problem states Let $R$ be the ring of $\mathbb{Z}_n$ of integers modulo $n$, where $n$ may ...
13
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3answers
1k views

A maximal ideal is always a prime ideal?

A maximal ideal is always a prime ideal, and the quotient ring is always a field. In general, not all prime ideals are maximal. 1 In $2\mathbb{Z}$, $4 \mathbb{Z} $ is a maximal ideal. ...
11
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5answers
477 views

How does a Class group measure the failure of Unique factorization?

I have been stuck with a severe problem from last few days. I have developed some intuition for my-self in understanding the class group, but I lost the track of it in my brain. So I am now facing a ...
10
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424 views

Simple example of non-arithmetic ring

Can anyone provide a simple concrete example of a non-arithmetic commutative and unitary ring (i.e., a commutative and unitary ring in which the lattice of ideals is non-distributive)?
4
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3answers
358 views

Norm and square of the ideal $(2,1+\sqrt{-5})$ in the ring of integers

Let $I=(2,1+\sqrt{-5})$ be an ideal of the ring of integers of $\mathbb Q(\sqrt{-5})$. What is its norm $N(I)$? And is $I^2$ principal? My notes say: $1$, $\sqrt{-5}$ is a $\mathbb Z$-basis ...
4
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2answers
146 views

Is there any non-monoid ring which has no maximal ideal?

Is there any non-monoid ring which has no maximal ideal? We know that every commutative ring has at least one maximal ideals -from Advanced Algebra 1 when we are study Modules that makes it as a very ...