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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48 views

Is $\Bbb{R}[X,Y]/(X^2+Y^2)$ a UFD or Noetherian?

Hello everyone I would like to know if $R$:= $\Bbb{R}[X,Y]/(X^2+Y^2)$ is UFD or Noetherian. I'm not really confortable in seeing how $\Bbb{R}[X,Y]/(X^2+Y^2)$ looks like. From what i've ...
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79 views

Which of the following is also an ideal?

If $U,V$ are ideals of a ring $R$, then which of the following is also an ideal of $R$? $U+V=\{u+v\mid u\in U,v\in V\}$ $U\cdot V=\{u\cdot v\mid u\in U,v\in V\}$ $U\cap V$ My attempt: I have ...
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27 views

Colon ideal and Artin-Rees lemma [on hold]

Let $a$ and $b$ be ideals of a Noetherian integral domain $R$. Prove that there exists a natural number $r$ such that $(a^n:b) = a^{n-r}(a^r:b)$ for $n > r$.
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14 views

A question from the Herstein's “noncommutative ring”, page 15, theorem 1.2.5

Let $R$ be a ring. $\rho$ is a maximal regular right ideal of $R$. $(R/\rho,+)$ is an irreducible $R$ module. $(r+\rho)\cdot r_1$ is defined by $rr_1+\rho$. $A$ is a two-sided ideal of $R$. ...
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1answer
50 views

If $p \in \operatorname{Ass}M$, then $R/P \subset M$.

Let $R$ be a commutative ring with unity. $M$ an $R$-module. Then $P \in \operatorname{Ass}M$ if and only if there is a submodule $N\subset M$ such that $R/P \cong N$. ...
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29 views

The relation of colon ideal and quotient [on hold]

Let $k$ be a field and $R=k[[x_1,x_2,\ldots,x_n]]$ the ring of power series over $k$. If $I$ is an ideal of $R$ such that the cardinal of the set $\{ann(f+I): f \in R \}$ is two, what can we say ...
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1answer
26 views

what does 2 ideals are equal mean?

I'm revisiting the proof of 1-1 correspondence theorem and while proving $f$ is one-one I don't know how to write mathematically what we mean by 2 ideals are equal? (Here $f$ is a map from set of ...
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4 views

Ideals of a skew polynomial ring where no positive power of the automorphism is inner

The exercise I'm trying to answer is as follows: Let $R$ be a ring, and $\alpha : R \rightarrow R$ an automorphism of $R$. Suppose that $R$ is simple and that no positive power of $\alpha$ is ...
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1answer
25 views

1-1 correspondence theorem

Here is the correspondence theorem stated as follows: Let $A$ be an Ideal of ring $R$.There is 1-1 correspondence between Ideals of $B$ containing $A$ and ideals of $R/A$. I have read the proof but ...
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44 views

Generators of an ideal in rings of power series

Please help me for solving a homework. Let $k$ be a field and $R=k[[x_1,x_2,\ldots,x_n]]$ the ring of power series over $k$. If $I$ is an ideal of $R$ such that ...
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35 views

How to find the number of maximal ideals? [on hold]

Let $n \geq 2$ and $n={p_1}^{e_1}{p_2}^{e_2}\cdots {p_r}^{e_r}$. Then the number of maximal ideal of $Z/nZ=Z_n$ is r n $e_1$+$e_2$+....+$e_r$ $p_1$$p_2$....$p_r$
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52 views

Relation between ideals in Noetherian domains.

Suppose that we have a Noetherian domain $R$ and two ideals $I$ and $J$ of $R.$ Now consider the minimal (or irredundant) primary decompositions $I=\bigcap\limits_{i=1}^r Q_i$ and ...
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+150

Determining consistency of a general overdetermined linear system

For $m > 2$, consider the $m \times 2$ (overdetermined) linear system $$A \mathbf{x} = \mathbf{b}$$ with (general) coefficients in a field $\mathbb{F}$; in components we write the system as ...
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2answers
87 views

Showing an ideal with maximality condition is prime.

Let $R$ be a commutative domain and suppose that $I \subseteq R$ is an ideal of $R$ maximal with respect to the property that $I^{-1} \not\subseteq R$. Show that $I$ is a prime ideal. This is ...
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20 views

Finding the element of the quotient ring $Z[i]/\langle 2+2i\rangle$

First, I'm writing an element to confirm whether I understood this quotient ring correctly. $$(5 + 7i) + \langle 2+2i \rangle = 2(2+2i) + (1+3i) + \langle 2+2i \rangle = ...
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45 views

For any two Ideals $A$ and $B$,$A+B=\langle A \cup B \rangle$

Below is the proof of : Prove that for any two ideals $A$ and $B$ of ring $R$,$A+B=\langle A \cup B~\rangle$ . Proof: By theorem (for any two ideals of a ring $R$ ,then the set $A+B$ is an ...
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0answers
21 views

Every right principal ideal non-emptily intersects the center — what is that?

This is a follow-up to Do Lipschitz/Hurwitz quaternions satisfy the Ore condition? Jyrki Lahtonen answered the question in the positive by noticing that every right principal ideal in either ring has ...
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26 views

How can we prove that every maximal ideal is a prime ideal? [closed]

In abstract algebra,how can we prove that every maximal ideal is a prime ideal? Give full logical proof.
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1answer
57 views

Sets of prime ideal contain a minimal element

I want to prove that every nonempty set of prime ideal contain a minimal element, my attempt is to prove it by using zorns lemma and i would like to know if my proof is valid. Let $\Sigma$ be a ...
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1answer
38 views

Colon ideal of fractional ideals is itself a fractional ideal

I received this question on homework in my homological algebra class and I need some guidance. Let $R$ be a commutative integral domain and $K$ be its field of fractions. A fractional ideal $I$ of ...
5
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1answer
68 views

How to show that $\mathbb{C}[x_1,x_2,x_3, x_4]/(x_1x_2 - x_4x_3, x_1x_3 - x_2x_4, x_4x_1 - x_3x_2)$ is integral domain

I am looking for a way to show that the ring $\mathbb{C}[x_1,x_2,x_3, x_4]/I$ where $I = (x_1x_2 - x_4x_3, x_1x_3 - x_2x_4, x_4x_1 - x_3x_2)$ is an integral domain. In other words I want to show $I$ ...
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39 views

Prove that, if $(a)=(a')$, then $a'=ua$

Let $R$ be integral domain. Show that if $2$ principal ideals $(a)$ and $(a')$ are equal (where $a,a'\in R$) then there exists $u\in R^{\times}$ such that, $a'=ua$ Now if $(a)=(a')$ then ...
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68 views

Radical ideals of $\mathbb{Z}$?

I am having trouble with classification of the radical ideals of $\mathbb{Z}$. We know that for a commutative ring $R$ with an ideal $I$, the radical of $I$ is defined (and denoted as $\sqrt{I}$) as ...
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30 views

Non Maximal Prime ideal! [duplicate]

Assume $S$ to be all continuous functions from $[0,1]$ to $\mathbb R$. I know by compactness of $[0,1]$ it follows that all maximal ideals of $S$ have the form $M_{x_0}=\{f\in S \mid f(x_0)=0\}$.Does ...
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2answers
57 views

Commutative ring and maximal ideal problem

Let $A$ be a commutative ring and $M$ be a proper maximal ideal in $A$. Show the following properties: (a) If each $a \in A \setminus M$ is a unit element in $A$, then $M$ is the only maximal ideal ...
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15 views

Improved Gröbner basis algorithm

I'm just learning about Gröbner bases and the Buchberger algorithm. I have seen chapters in several pieces of literature that deal with improving the Buchberger algorithm, but they never seem to ...
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19 views

a question on equivalence classes of balanced fractional ideals and Dedekind domain

Let $R$ be a commutative ring, and let $K=R\otimes \mathbb{Q}$. Def.1) We say that a pair of fractional ideals $(I, I')$ in $K$ is balanced if $II'\subseteq R$ and $N(I)N(I')=1$. Def.2) Two ...
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1answer
52 views

Preimage of maximal ideal is maximal [duplicate]

I've just started a commutative algebra course and I'm stuck on the very first homework problem: Let $A \not= \{0\}$ be a commutative ring. Let $\Phi : A \longrightarrow B$ be a ring homomorphism ...
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1answer
65 views

Intersection of two polynomial ideals

In the $4$-dimensional affine space $\mathbb{A}^4$ with coordinates $x,y,z,t$, consider $X$ as the union of the planes $$ X'=\{x=y=0\} $$ and $$ X''=\{z=x-t=0\} $$ (I'm working on a algebraically ...
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116 views

Show that every maximal ideal in $ \mathbb{Z}[x, y] $ contains a prime number [closed]

Let $\mathfrak{M} \subseteq \mathbb{Z}[x, y]$ be a maximal ideal. Show that $ \exists\ p \in \mathbb{Z}$, $p$ prime such $p \in \mathfrak{M}.$ Thanks for the answers. I'd be interested in a proof ...
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1answer
28 views

Let $I=(2,X)$ and $J=(3,X)$ be ideals of $\mathbb{Z}[X]$. Prove $V=\{i \cdot j : i \in I, j \in J\}$ is not an ideal. [closed]

Let $I=(2,X)$ and $J=(3,X)$ be ideals of $\mathbb{Z}[X]$. Prove $V=\{i \cdot j : i \in I, j \in J\}$ is not an ideal. Possible strategy: find $v_1,v_2 \in V$ such that $v_1+v_2 \notin V$.
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An error in the book “noncommutative ring” writed by Herstein

I'm reading the book "noncommutative ring" writed by Herstein. In the page 15, the author says that Let $F$ be a field and $A$ is an algebra over $F$. Let $\rho$ be a maximal regular right ideal ...
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1answer
50 views

Find all the ideals of $\mathbb Q[X]$

I am trying to find all the ideals of the ring $\mathbb Q[X]$. If $I$ is a non trivial ideal of $\mathbb Q[X]$, then there exists $p(x) \in \mathbb Q[X]$. Since $I$ is an ideal and a group under ...
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2answers
77 views

Are ideals necessarily definable?

Consider the first-order language of rings. Let $R$ be a ring and $I \subseteq R$ be an ideal. Is $I$ necessarily $\emptyset$-definable? If not, what if we allow parameters from $R$?
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1answer
33 views

Left ideals of $M_n(K)$ [duplicate]

Let $K$ be a field and $n \in \mathbb N$. Show the following (i) Let $V \subset K^n$ be a subspace and $I_V$ the subset of $M_n(K)$ consisting of all the matrices whose rows belong to $V$. Prove that ...
2
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1answer
31 views

Equality of ideals and their vareties.

Let $I_1 $,$I_2 $ $\in \mathbb{C}[x_1,x_2,...,x_n] $ be two polynomial ideals. If their affine varieties, $\mathbb{V}(I_1)=\mathbb{V}(I_2)$ are equal then is $I_1=I_2$ always?
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1answer
79 views

non-principal height one primes of a particular hypersurface

I was reading about divisor class groups, and I was wondering the following. Let $R=\mathbb{C}[X,Y,Z,W]/(XZ-YW)$, and let $x,y,z,w$ be the images of $X,Y,Z,W$ in $R$, respectively. Is there a way ...
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1answer
41 views

Problem on the number of generators

I have got stuck with two generator problems: The ideal $(zx,xy,yz)$ can't be generated by $2$ elements The ideal $(xz-y^2,yz-x^3,z^2-xy)$ can't be generated by $2$ elements Here the ...
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4answers
166 views

A finitely dimensional algebra over a field has only finitely many prime ideals all of them are maximal

Let $K$ be a field and let $R$ be a $K$-algebra with unity which is finite dimensional as a $K$-vector space. Prove that $R$ has only finitely many prime ideals all of which are maximal. (Hint: ...
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1answer
38 views

Product of two non-principal ideals

I have problems understanding why $$(6,2+\sqrt{-56})(6,-2+\sqrt{-56})=6(2,\sqrt{-56})$$ in $\mathbb{Z}[\sqrt{-14}]$. By definition the product of two ideals $$IJ=\sum_{i,j}^{k}f_{i}g_{j}$$ ...
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1answer
87 views

Analogy of ideals with Normal subgroups in groups.

I've started with Ideals in ring theory but still not comfortable with the analogy it has with normal subgroups in group theory.Like we can visualize normal subgroups as Is there some good intutive ...
3
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1answer
77 views

Are there any commutative rings in which no nonzero prime ideal is finitely generated?

Are there any commutative rings in which no nonzero prime ideal is finitely generated? I feel like the example (or proof of impossibility) ought to be obvious, but I'm not seeing it.
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0answers
17 views

Size of a subset of the set of units of a quotient ring

Let $R$ be a commutative Dedekind domain with multiplicative identity $1$, let $k$ be a positive integer, and let $I$ be a nonzero proper prime ideal of $R$. Is there a way to find the size of the set ...
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2answers
65 views

Find a generator for an ideal in $\mathbb{Q}[T]$

Let $I$ be the ideal in $\mathbb{Q}[T]$ generated by $L=\{T^{2}-1, T^3-T^2+T-1,T^4-T^3+T-1\}$. Find $f\in\mathbb{Q}[T]$ such as $(f)=f\mathbb{Q}[T]=I$. The book solution proves that $I\subseteq ...
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1answer
80 views

Ideal generated by a regular sequence

I need to prove that the ideal $$ I = (xz -y^2, x^2t^2 -yz^3, x^2yt^2 -xz^4) \subset R = \mathbb{K}[x,y,z,t]$$ is generated by a $R$-regular sequence. How can I do it? I don't know if this can ...
4
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1answer
45 views

Does there exist an ideal in $\mathbb{Z}_4[x]$ which is prime but not maximal?

Question: Does there exist an ideal in $\mathbb{Z}_4[x]$ which is prime but not maximal? Thoughts: It seems to me that the ideal $(x)$ fails to be a prime ideal since $0 \in (x)= 2 * 2$ with $2 ...
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3answers
60 views

Prove that $I$ is a maximal ideal of $\mathcal A$. [duplicate]

Please, give-me a hint to prove this proposition: Let $\mathcal A$ be the ring of all continuous real functions (with the usual operations of sum and multiplication) defined on the interval ...
4
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0answers
75 views

Functorial approach to Ideals and Quotients, Multiplicative Sets and Localizations

I have been playing with substructures of commutative rings today and noticed that there is a strong analogy between the formation of quotients and kernels with the formation of localizations with ...
1
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1answer
35 views

Ideal quotient and extension

Let $R$ be a commutative ring and $S$ a subring of $R$. If $I$ is an ideal of $S$ define $I^e$ as the ideal in $R$ generated by $I$, i.e. the extension of $I$ in $R$. If $I,J$ are ideals in $S$, we ...
7
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3answers
212 views

In $\mathbb{Z}/(n)$, does $(a) = (b)$ imply that $a$ and $b$ are associates?

[Update: Based on the hints provided by @zcn and @whacka, I believe I have found a solution. See my answer below.] Below, $R$ is a commutative ring with $1$. In John J. Watkins' Topics in ...