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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Finiteness of ideal of given norm

I'm trying to prove that there are only finitely many ideals of any given norm in the ring of integers $\mathcal{O}_k$ over a numberfield $K$. I know there are "standard proofs" (eg How many elements ...
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341 views

Ideal generated by a set in a commutative ring without unity

In a commutative ring with unity $1$, call it $R$, the the ideal generated by the set $S=\{a_1,...,a_n\}$ is the smallest ideal of $R$ containing $S$. It can be proven that this ideal is $$ ...
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1answer
65 views

The set of zero divisors is the union of radicals of annihilators

I am trying to figure out why the statement $$\text{the set of zero divisors }=\bigcup_{0\ne x\in R} \sqrt{\text{Ann}(x)}$$ is true. Here $R$ is a commutative ring, $\text{Ann}(x)=\{r\in R\mid rx=0\}$ ...
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40 views

If a polynomial ideal can be generated by $k$ elements, can it be generated by $k$ elements of any generating set?

Let $I = (p_1,\ldots, p_k) \subset \mathbb{C}[x_1,\ldots,x_n]$. If we have a set of $k'$ polynomials such that $(q_1,\ldots,q_{k'}) = I$, can we always find a $k$-member subset such that ...
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94 views

Principal Ideal Domain

Let $D$ be a principal ideal domain and let a be some fixed element of $D$. Let $(a)$ denote the ideal generated by $a$. Prove that if $a$ is irreducible and $I$ is an ideal of $D$ such that ...
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How to recover the ideal from grobner basis of kernel of ann(x)

M -> ann(x) i can find the grobner basis of kernel of ann(x) and need the final step to recover this basis to ideal as i know, eliminate is not for all cases, what is the general practice to treat ...
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How to show an ideal is zero-dimensional? [duplicate]

Let $J$ denote the ideal in $\mathbb{Q}[x,y,z]$ generated by $\{y^2-xy-2xz,y^3+z^2+1, x^2yz-yz\}$. Show that $J$ is zero-dimensional. How do I go about showing this?
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167 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 ...
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200 views

Krull Dimension

I'm studying Krull dimension and I'm confused about the definition of $\text{ht}(P)$, which is as I understand is the following: let $$P_0\subset P_1\subset\dots\subset P_n=P$$ be a chain of prime ...
2
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74 views

Union of Associated Primes being finite.

Let $R$ be a commutative Noetherian ring with unit. Let $I=(x_1,x_2,\dots,x_t)$ be a nonzero ideal of $R$. Define $I_n=(x_1^n, x_2^n,...,x_t^n)$. Are there known results about $\cup_n ...
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421 views

Help with proof that $\mathbb Z[i]/\langle 1 - i \rangle$ is a field.

I have been having a lot of trouble teaching myself rings, so much so that even "simple" proofs are really difficult for me. I think I am finally starting to get it, but just to be sure could some one ...
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4answers
310 views

Help with proof that $I = \langle 2 + 2i \rangle$ is not a prime ideal of $Z[i]$

(Note: $Z[i] = \{a + bi\ |\ a,b\in Z \}$) This is what I have so far. Proof: If $I$ is a prime ideal of $Z[i]$ then $Z[i]/I$ must also be an integral domain. Now (I think this next step is right, ...
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88 views

Proving a factorization of ideals in a Dedekind Domain

Let $R=\mathbb{Z}[\sqrt{-13}]$. Let $p$ be a prime integer, $p\neq 2,13$ and suppose that $p$ divides an integer of the form $a^2+13b^2$, where $a$ and $b$ are in $\mathbb{Z}$ and are coprime. Let ...
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2answers
65 views

$(ax)(ay) = a(xy) \in (a)$

Lemma 26.1 Let $R$ be a commutative ring with unity element $e$. The set $(a) = \{ar : r \in R\}$ is an ideal of $R$. Proof. First, we will show that $(a)$ is a subring of $R.$ Since $a = ae$ then $a ...
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124 views

For a ring R, and ideals $A$, $B$, then $AB=A \cap B$ if $A + B = R$

$AB \subseteq A \cap B$ is clear. I have seen reverse inclusion proven thus, Let $x \in A\cap B$. Since $A+B=R$, there exist $a \in A$, $b \in B$, such that $a+b=1$. Then $x= axa + axb + bxa + bxb$. ...
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122 views

Factoring the ideal $(8)$ into a product of prime ideals in $\mathbb{Q}(\sqrt{-7})$

I am trying to factor the ideal $(8)$ into a product of prime ideals in $\mathbb{Q}(\sqrt{-7})$. I am not exactly sure how to go about doing this, and I feel I am missing some theory in the ...
4
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1answer
149 views

Solving $x^2+19=y^5$

I was given several exercises and there is a particular one, I am not able to solve. Let it be given that $Pic(\mathbb{Z}[\sqrt{−19}])$ is a finite group of order $3$. Use this to find all integral ...
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68 views

Maximal Ideals of the Wiener Algebra

I'm wondering why the maximal ideals of the Wiener algebra $\mathcal{W}$ are of the form $\{M_z:z\in \mathbb{T}\}$ where $M_z=\{f\in \mathcal{W}\; |\; f(z)=0\}$. Given that the Wiener algebra is a ...
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1answer
96 views

Integral extension implies that the induced map on prime spectra is closed

Say we have an integral extension $f:R \hookrightarrow S$ of rings. I want to show that the induced map $f^*:Spec(S) \twoheadrightarrow Spec(R)$ is closed. In other words, let $V(I) = \{\mathfrak{P} ...
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votes
3answers
227 views

Let $R$ be a finite commutative ring. Show that an ideal is maximal if and only if it is prime.

Let $R$ be a finite commutative ring. Show that an ideal is maximal if and only if it is prime. My attempt: Let $I$ be an ideal of $R$. Then we have $I$ is maximal $\Leftrightarrow$ $R/I$ is a finite ...
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Ideals of the algebra of all bounded linear operators on $\ell_p \oplus \ell_q$

Let $\mathcal{L}(X)$ be the algebra of all bounded linear operators from $X$ to $X$ for Banach space $X$. I need to show that $\mathcal{L}(\ell_p \oplus \ell_q)$ for $p \neq q$ contains at least two ...
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primary decomposition of ideals

How to find the primary decomposition of ideal $I = (X^2, XY, XZ, YZ)$ in the ring $k[X,Y,Z]$? Also how to show that $(X,Y)^{308}$ is primary ideal in $k[X,Y,Z]$? Is there a general rule for finding ...
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121 views

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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217 views

Radical/Prime/Maximal ideals under quotient maps

Let $I$ be an ideal of a ring (commutative with unity) $R$ and let $q:R\to R/I$ be the quotient map. Then there is a well known correspondence between ideals of $R$ containing $I$ and ideals of $R/I.$ ...
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1answer
90 views

Ideal commutative rings

Let $R$ a commutative ring and $I$, $J$ ideals of $R$ such that $I + J = R$. Prove that $IJ = I \cap J $ Is clear that $IJ \subseteq I$ and $IJ \subseteq J$ then $IJ \subseteq I \cap J$ this for ...
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117 views

true or false questions on prime and maximal ideals

which of the following are true and which are false:- $1.$ Every prime ideal of every commutative ring with unity is a maximal ideal. $2.$ The prime subfield of $\mathbb{C}$ is $\mathbb{C}$ $3.$ ...
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331 views

Local ring $R_p$ no nilpotent elements, then $R$ no nilpotent elements

Question: Let $R$ be a ring. Suppose that for every prime ideal $p \lhd R$ the local ring $R_p$ ($=(R\setminus p)^{-1}R$) has no non-zero nilpotent elements. Prove that $R$ has no non-zero ...
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Factorization of $5$ in the splitting field of $x^3 + 2$

I wonder if someone could help to clarify the following. Let $\zeta$ be a primitive cube root of unity and $\alpha = \sqrt[3]{2}$. Let $K = \mathbb{Q}(\alpha)$ and $L = K(\zeta)$. Then $L$ is the ...
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2answers
55 views

Let $\mathfrak{m}$ be maximal in $R$. Show that $\mathfrak{m}^n/\mathfrak{m}^{n+1}$ is an $R/\mathfrak{m}$-vector space for all $n\geq 0$

Let $R$ be a commutative ring and $\mathfrak{m}$ a maximal ideal of $R$. Show that $\mathfrak{m}^n/\mathfrak{m}^{n+1}$ is an $R/\mathfrak{m}$-vector space for all $n\geq 0$ I mostly just want to ...
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139 views

Show that $\mathbb{Z}[x]=\lbrace \sum_{i=0}^{n}{a_ix^i}:a_i \in \mathbb{Z}, n \geq 0 \rbrace$ is not a principal ideal ring.

Show that $\mathbb{Z}[x]=\lbrace \sum_{i=0}^{n}{a_ix^i}:a_i \in \mathbb{Z}, n \geq 0 \rbrace$ is not a principal ideal ring. I know the definition of principal ideal ring is that every ideal is ...
2
votes
1answer
71 views

ideals in $C^*$ algebra

Let $A$ be a $C^*$ algebra and $I$ be a closed ideal in $A$. Prove that for all $a\in A$, $a\in I$ iff $a^*a\in I$. I want to prove that if $a^*a\in I$, then $a\in I$, and I know the following fact ...
4
votes
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225 views

Definition of principal ideal

This is a pretty basic question about principal ideals - on page 197 of Katznelson's A (Terse) Introduction to Linear Algebra, it says: Assume that $\mathcal{R}$ has an identity element. For $g\in ...
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1answer
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Ideal gen. by a set S = Intersection over ideals containing S

I am trying to prove the following statement: Let R be a ring and $I=\{\sum_{i=1}^n a_i x_i : a_i\in R\}$ the ideal generated by $S=\{x_1,\ldots, x_n\}$. Then $I$ is the intersection of ideals $J$ in ...
3
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0answers
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Proof about affine varieties

Ok so I have that $k$ is algebraically closed and $F$ is an element of $k^n$, and it is a reduced polynomial. We have that $V = V(F)$. In the book it says prove that $F$ generates $I(V)$ but in my ...
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votes
1answer
140 views

$I\cap J = P$ prime ideal, then $P=I$ or $P=J$

Question: Prove that if $I,J$ are ideals and $I\cap J=P$ is a prime ideal, then either $P=I$ or $P=J$. My proof: Suppose $P\ne I$. Then $I\cap J=P\subsetneq I$ and $\exists i\in I\setminus P$. Now ...
3
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1answer
60 views

Finding specific ideals of a ring

How does one go about finding all the ideals of a ring (quadratic extension) containing a specific number? In particular find all the ideals of $\mathbb Z(\sqrt{-29})$ which contain the integer 30.
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1answer
178 views

Ideal in compact Hausdorff space

This is exercise 70, chapter 4. from Folland (page 142) Let $X$ be a compact Hausdorff space. An ideal in $C(X, \mathbb{R})$ is a subalgebra $J$ of $C(X, \mathbb{R})$ such that if $f\in J$ and $g\in ...
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Affine variety over a field

Suppose we have an algebraically closed field $K$. An affine variety is the common zero locus of a collection of polynomials $f_{\alpha} \in K[z_1, \dots, z_n]$. So basically it is the set of points ...
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1answer
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Hilbert's Weak Nullstellensatz Variety Ideal

I have the following question.... $f=6x^2y-xy^2-2y^3+1\ and \ h=3x-2y \in \mathbb{C}[x,y] $ Im asked to Show that V(f,h) is empty.. But im not sure what method I use to show this... Then im ...
3
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1answer
112 views

Gröbner Basis for Ideal $J$

I have the following question... Consider the ideal $J:= (x^2y-x^2y^2,\ x^2z-z^2yx,\ x^2+xz) \subset \mathbb{Q}[x,y,z]$ Is $x \in J?$ Is $x \in \sqrt{J} $? I know finding if $x$ is in the radical ...
4
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Where do I use the fact that $F$ is algebraically closed in this proof?

I have to do the following. Let $F$ be an algebraically closed field. $I\in F[X_1,...,X_n]$ an ideal. Denote by $S(I)$ the subset in $F^n$ consisting of all $n$-tuples $(a_1,...,a_n)\in F^n$ such that ...
7
votes
1answer
129 views

Ideal of ideal needs not to be an ideal

Suppose I is an ideal of a ring R and J is an ideal of I, is there any counter example showing J need not to be an ideal of R? The hint given in the book is to consider polynomial ring with ...
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votes
3answers
116 views

Find ideals of ring

I am stuck with a homework problem. Let $R=\mathbb{Z}[\sqrt{ -3}]$. a) Find an ideal $I$ of $R$ such that $(4) \subsetneq I \subsetneq R$. Explain why the inclusions $\subsetneq$ in my example are ...
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Subset of a P-ideal need not be a P-ideal

I was looking for examples showing that subset of a P-ideal is not necessary. I will post below a counterexample I was able to find. (I hope it is correct.) But I'd be glad to see other simple (or ...
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3answers
540 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 $e \in A$ ...
3
votes
1answer
104 views

Computing kernel of ring homomorphism

I am trying to answer the question already asked here. My question is two parts: 1) I think I have found a proof on my own, could someone check it is valid? Modulo that ideal, $x_i\equiv a_i$ so ...
3
votes
1answer
190 views

Proof about Noetherian rings

I have to prove that every finite ring is Noetherian. I know examples of Noetherian rings which are not finite such as the field of complex numbers or a PIR like the integers. But anyway: [Proof]: I ...
2
votes
2answers
78 views

$ I(J+L)=IJ+IL$ if $I,J,L$ are ideals of $K$

Given that $I,J,L$ are ideals of $K$, do we have $I(J+L)=IJ+IL$? I am confused how to do it.
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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 ...
9
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1answer
225 views

Conditions for $\sqrt{\mathfrak{a + b}} = \sqrt{\mathfrak{a}} + \sqrt{\mathfrak{b}}$

Let $A$ be a commutative ring with identity and, $\mathfrak{a}$ and $\mathfrak{b}$ ideals.I'm trying to find sufficient and necessary conditions for $\sqrt{\mathfrak{a + b}} = \sqrt{\mathfrak{a}} + ...