Tagged Questions

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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Maximal and prime ideals of quaternions with integer coefficients

Let $R = \mathbb{Z} + \mathbb{Z}i + \mathbb{Z}j + \mathbb{Z}k$, the subring of $\mathbb{H}$ consisting of quaternions with integer coefficients. An exercise in Goodearl and Warfield's An Introduction ...
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Is $I=\langle7, 3+\sqrt{19}\rangle$ a principal ideal of $\Bbb Z[\sqrt{19}]$?

Is $I=\langle7, 3+\sqrt{19}\rangle$ a principal ideal of $\Bbb Z[\sqrt{19}]$? I defined the norm : $$N(a+b\sqrt{19})=(a+b\sqrt{19})(a-b\sqrt{19})=a^2-19b^2$$ Then we can see the multiplicative ...
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12 views

Show that $\mathbb{Z}[\sqrt{223}]$ has three ideal classes.

Well the question is the title. I tried to grab at some straws and computed the Minkowski bound. I found 19,01... It gives me 8 primes to look at. I get $2R = (2, 1 + \sqrt{223})^2$ $3R = (3, 1 + ...
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11 views

Proving Irreduciblity in Polynomial Quotient Rings

I'm working on an exercise from Dummit and Foote, and I've gotten down to the following lemma that makes everything I need work out, the only problem is that I'm not sure how to prove it (or whether ...
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14 views

Does every algebra automorphism preserve augmentation ideal filtration?

Let $A=\displaystyle\bigoplus_{n\geq0}A_n$ be a graded algebra and let the augmentation homomorphism $\varepsilon:A\to A_0$ be the projection. Define the augmentation ideal, denoted by $A_+$, to be ...
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16 views

Simple Maximal Ideal Question.

Question: Let $R = \{a + bi | a,b \in \Bbb Z \}$, let $M = \{x(2 + i) | x \in R\}$. Prove M is a maximal ideal of R. I just started learning about ideals so I apologize for asking a basic question, ...
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22 views

Let $I$ be a proper ideal of a ring $R$. Then $IR[\alpha_1, … , \alpha_n]$ is a proper ideal of $R[\alpha_1, … , \alpha_n]$

Let $I$ be a proper ideal of the commutative ring $R$. Then $IR[\alpha_1, ... , \alpha_n]$ is a proper ideal of $R[\alpha_1, ... , \alpha_n]$ I thought of using the fact that an ideal of any ring ...
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20 views

Is the set of pseudo-complements of the elements of an ideal in a pseudocomplemented lattice a filter?

Let $L$ be a pseudocomplemented distributive lattice with $0$ and $1$, $I \subseteq L$ an ideal and set $F = \{\neg x \; | \; x \in I\}$, where $\neg x$ is the pseudocomplement of $x$. My question is: ...
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15 views

Number Theory: Ramification

I am currently trying to figure out the following question regarding ramification. Let K = $\mathbb{Q}(\sqrt{5})$, L = $\mathbb{Q}(\sqrt{7})$, M = $\mathbb{Q}(\sqrt{35})$, and KL = ...
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Are there irreducible ideals that are not primary in $K[X_1,\dots,X_n,\dots]$?

I can give examples of non-noetherian rings having irreducible ideals that are not primary. Among them are idealizations and valuation domains. But the first non-noetherian ring we are thinking about ...
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42 views

Why are these two quotients equal?

I'm not being able to check why are these two quotients equal. $\mathbb C[x]/(x^2-x^3)= \mathbb C[x]/(x^2)$ Can someone tell me why is it valid?
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1answer
23 views

Ideals in a polynomial ring over a skew field

I know that a polynomial ring over a field is a PID, does this property also hold for a polynomial ring over a skew field? Is there maybe something else that characterise the ideals in that ring ?
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1answer
32 views

An equivalent condition with $\{0\}$ being the only nilpotent ideal

In a ring $R$ prove that $\{0\}$ is the only nilpotent ideal if and only if for every ideals $A$ and $B$ from $R$, $AB=\{0\}$ implies $A\cap B=\{0\} $.
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21 views

Are prime ideals always comaximal?

This is easy to see in the ring of integers. In fact, the ideals don't even have to be prime. It's enough to be coprime. Then their GCD is 1, so 1 can be written as a linear combination of the ...
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36 views

The ideal $(x_1,x_2,x_3,…)$ (with infinitely many variables) in the ring $K[x_1,x_2,x_3,…]$ is not finitely generated. [duplicate]

How can I show that the ideal $(x_1,x_2,x_3,...)$ (with infinitely many variables) in the ring $K[x_1,x_2,x_3,...]$ is not finitely generated. I cannot complete the arguments as I thought if the ...
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49 views

Primary decomposition of $(x^2,xy,xz)$ in $k[x,y,z]$ where $k$ is a field

I am looking for the primary decomposition of $(x^2,xy,xz)$ in $k[x,y,z]$ where $k$ is a field. I am not looking for a solution here, rather a hint or two. Is there a general strategy for approaching ...
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96 views

If a module is nonzero, then a localization module is nonzero

Let $R$ be a commutative ring, when $\mathfrak p$ is a prime ideal, there is the localization $M_{\mathfrak p}:=S^{-1}M$, where $S=R\setminus\mathfrak p$. Show: If $M$ is a nonzero $R$-module, ...
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1answer
20 views

How to directly show that $\mathbb{Z}_{(p)}$ is a local ring with the unique maximal ideal $p \mathbb{Z}_{(p)}$?

I know that $\mathbb{Z}_{(p)}$ is a local ring because it's the localization of $\mathbb{Z}$ over $p$, but is there a direct way to prove that and find its unique maximal ideal? I've been ...
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31 views

A concrete example of a unital noncommutative ring without maximal two-sided ideals

Whenever I say ideal in this question I'm talking about two sided-ideals. Does there exist a concrete example of a non-commutative ring with $1$ without maximal ideals? We know that if $R$ is a ...
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1answer
19 views

Let $R=\mathbb{Z}_2[X]$ and $I=<x^2+1,x^3+1>$ then which of the following are in $I$?

Let $R=\mathbb{Z}_2[X]$ and $I=<x^2+1,x^3+1>$ then which of the following are in $I$? $x^4+1,x^5+x+1,x^6+1$ $(x^2+1)^2=x^4+1,(x^3+1)^2=x^6+1$ About second option I need help.
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Question about the wording of a Ring Theory problem involving ideals

The homework question is: If $I,J$ are ideals of $R$, let $IJ$ be the set of all sums of elements of the form $ij$, where $i \in I$, $j \in J$. Prove that $IJ$ is an ideal of $R$. The phrase "the ...
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26 views

Which one of the following ideals is radical?

For a commutative ring $A$ and an ideal $I$, $N(I)=\{x\in A\mid x^n\in I \ \mbox{for some integer}\ n\}$. Then which of these satisfy $N(I)=I$: $A=\mathbb{Z}, I=(2)$, $A=\mathbb{Z}[x], ...
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26 views

How to prove a subset is an ideal

I just started learning about Ring Theory today and I am having some trouble truly understanding and being able to apply certain concepts. The first concept I am having trouble understanding is an ...
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3answers
65 views

Rigorous proof that certain ideal is not finitely generated

Let $R = F[x_1,x_2,\ldots]$ (polynomials in an infinite number of indeterminates) and let $I = \{f \in R : f(0,0,\ldots) = 0\}$. One can easily see that this is indeed an ideal. The proof for why ...
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1answer
47 views

The ideal $\langle x,y \rangle$ in $F[x,y]$ is not principal. [duplicate]

Let $F$ be a field. Apparently we know that $\langle x,y \rangle \neq \langle g(x,y) \rangle$ for any $g \in F[x,y]$. Why is this the case?
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1answer
54 views

Operations with ideals in a commutative ring

Let $R$ be a commutative ring with identity. Let $A$ and $B$ be ideals in the ring $R$. It is true that $(A\cap B)(A+B)$ equals the product $AB$?
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44 views

$\mathfrak{a}_{1} + \dots + \mathfrak{a}_{n} = A \Rightarrow \mathfrak{a}_{1}^{r_{1}} + \dots + \mathfrak{a}_{n}^{r_{n}} = A$

I have to prove the following : Let $A$ be a commutative ring with unity and let $\mathfrak{a}_{i}$ be ideals in $A$. Assume that $\mathfrak{a}_{1} + \dots + \mathfrak{a}_{n} = A$. Let $r_{i}$ be ...
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4answers
117 views

Is product of prime ideals prime?

I'm trying to show that the product of ideals $(x_1, x_3)$ and $(x_2, x_4)$ in $\mathbb C[x_1, x_2, x_3, x_4]$ is a radical ideal, but no other way that I can think of works. So, is the product ...
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1answer
117 views

Questions related to maximal ideals

In my previous sessional exams, I was asked to prove these two: 1) Find a ring which doesn't have a maximal Ideal. 2) If a ring has unity, then it has a maximal Ideal. About the ...
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18 views

Verifying proof :an Ideal $P$ is prime Ideal if $R/P$ is an integral domain.

I had to write the proof to show that an Ideal $P$ of a commutative ring $R$ is prime Ideal if $R/P$ is an integral domain. let $a,b\in R$ s.t. $ab\in P$ , ...
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1answer
37 views

Showing that an ideal is prime

I think that $k[x,y,z]/(z-1, x^2-y)$ can be identified as a subset of $k[x,y]$ with all polynomials whose $x$ terms are only degree one. Therefore I conclude that $k[x,y,z]/(z-1, x^2-y)$ is integral ...
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1answer
41 views

no. of elements in $\mathbb Z[i]/\langle 3+i\rangle$. [duplicate]

$\mathbb Z[i]/\langle 3+i\rangle$ can be represented as :$\{a+3b+\langle 3+i\rangle\big|~~a,b\in \mathbb Z\}$ How shall I find the total no. of elements in $\mathbb Z[i]/\langle 3+i\rangle$.. ...
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1answer
21 views

$A=\{(3x,y)\mid x,y\in \mathbb Z\}$ is a maximal ideal of $\mathbb Z \oplus \mathbb Z$

I've a question from gallian which states: Show that $A=\{(3x,y)\mid x,y\in \mathbb Z\}$ is a maximal ideal of $\mathbb Z \oplus \mathbb Z$.Generalize.What happens if $3x$ is replaced by $4x$... ...
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1answer
35 views

Doubt regarding zero elements in factor ring :$\mathbb Z[i]/\langle3-i\rangle$

I have the factor ring $\mathbb Z[i]/\langle3-i\rangle$ and am asked to find elements zero in this ,they are $0,3-i,i(3-i),(3-i)+i(3-i)$. But I can't understand how do we guarantee these are the only ...
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29 views

Ideal in power series ring

Let $J$ be an ideal in $k[[x_1,...,x_n]]$ such that $(x_{1},...,x_{n})^{2}\subseteq J$, $\{x_{1},...,x_{r}\}\nsubseteq J$ and $\{x_{r+1},...,x_{n}\} \subseteq J$, for some $1\leq r \leq n$. I want to ...
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1answer
22 views

An Ideal which is Maximal additive subgroup is a Maximal Ideal

How should I prove this: Any Ideal which is a Maximal additive subgroup is also a Maximal Ideal . any idea how to prove it..
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1answer
27 views

Factorization in Dedekind domains

Let $R$ be a commutative, Dedekind (and therefore Noetherian) ring with $1$. Let $I$ be a non-prime ideal of $R$, and let $a,b$ be elements of $R$ such that $a\not\in I,b\not\in I$ but $ab\in I$. Let ...
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37 views

An ideal which is not finitely generated

Let $K$ be a field and $A=K[x_1,x_2,x_3,...]$. Prove that the ideal $I:=\langle x_i: i \in \mathbb N\rangle$ is not finitely generated as $A$-module. I have no idea what can I do here, I mean, ...
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1answer
34 views

A question regarding the number of generators of an ideal [duplicate]

Let $I$ be an ideal in $\mathbb{C}[x_1 ,x_2 ,x_3 ,x_4 ]$ such that $I$ is generated by $x_1 x_3$, $x_2 x_3$, $x_1 x_4$, and $x_2 x_4$. How to show that this I cannot be generated by two ...
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2answers
45 views

Showing that an ideal is maximal

Let $k$ be an algebraically closed field and $f$ be the polynomial $x_1x_2+x_2x_3+x_3x_1$ in $k[x_1, x_2, x_3]$. Here $f$ is irreducible. Then this polynomial ring is not a $PID$, it is only an ...
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13 views

Ideal from ring of fraction

Given $R$ is a commutative ring with $1$ and $D$ is multiplicatively closed containing $1$, I want to show that any ideal of $D^{-1}R$ is of the form $D^{-1}I$, where $I$ is an ideal in $R$. I have ...
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1answer
47 views

$\overline{\mathbb{Z}}$ is not a Dedekind domain.

I have to prove the following statement : Let $\overline{\mathbb{Z}}$ be the ring of all algebraic integers in (a fixed choice of) $\overline{\mathbb{Q}}$. Then $\overline{\mathbb{Z}}$ is not a ...
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Ideal and its number of generators [duplicate]

Consider an ideal $I$ in $\mathbb{C}[x_1, x_2, x_3, x_4]$ such that $I$ is generated by $x_1x_3, x_2x_3, x_1x_4,$ and $x_2x_4$. I think this ideal cannot be generated by two elements, but can't ...
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1answer
26 views

If $\cap_{j=1}^{n}I_{j} \subseteq P$ for any ideals $I_1,I_2,..I_n$, then $I_j \subseteq P$ for some $j$

$P$ is a prime ideal if $P$ satisfies the following : If $\bigcap\limits_{j=1}^{n}I_{j} \subseteq P$ for any ideals $I_1,I_2,..I_n$, then $I_j \subseteq P$ for some $j$, where $R$ is a commutative ...
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2answers
55 views

Sum and product of comaximal ideals

Let $R$ be a commutative ring with unity. If $R=I_{i}+I_{j}$, for all $i\ne j$, where $I_1,I_2,...,I_n$ are ideals of $R$, I want to show that $$R=I_{n}+I_{1}I_{2}\cdots I_{n-1}.$$ I started off ...
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26 views

Ideals of a field

I had the following - apparently straightforward - question on one of my past assignments: Show that a field has no other ideals except $\{0\}$ and the field itself. This was the proof I gave: ...
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1answer
19 views

What exactly does it mean for a maximal ideal to be unique in a principal ideal domain?

I'm currently reading about PIDs and have come across a question involving maximal ideals which at one point reads "Suppose that a Euclidean domain $R$ had a unique maxima ideal $P$". Does this mean ...
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30 views

Prove that the intersection of all maximal left ideals of a ring $R$ is a two sided ideal

Prove that the intersection of all maximal left ideals of a ring $R$ is a two sided ideal. What i did:Suppose $B$ be the intersection of all maximal left ideals of the ring $R$. Clearly $B$ is a left ...
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2answers
149 views

How to show that there is a bijective correspondence between two sets of prime ideals

I'm trying to solve this Algebra Problem, and I'm not quite sure, if I'm on the right way. Let $R$ be a commutative ring and $S \subset R$ a multiplicative subset. Show that $p \to pS^{-1}R$ ...
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1answer
32 views

Localizing at maximal ideals and the product

Let $D$ be an integral domain, $M_{i}$, $i = 1,...,r$ be some of its mutually distinct maximal ideals, and $e_{i}$be positive integers for all $i$. Is it true in general that the extension of the ...