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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144 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
votes
1answer
168 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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0answers
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 ...
1
vote
1answer
111 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} ...
9
votes
3answers
315 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 ...
12
votes
1answer
132 views

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 ...
4
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0answers
150 views

Primary decomposition of ideals

How to find a primary decomposition of the ideal $I = (X^2, XY, XZ, YZ)$ in the ring $k[X,Y,Z]$? Is there a general rule for finding primary decompositions? Also how to show that $(X,Y)^{308}$ is a ...
1
vote
3answers
140 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 ...
3
votes
1answer
265 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 $R/I.$ ...
1
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1answer
103 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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1answer
134 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 $\mathbb{C}$ $3.$ ...
1
vote
1answer
377 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 ...
6
votes
1answer
109 views

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 ...
0
votes
2answers
58 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 ...
5
votes
3answers
143 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
73 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
3answers
251 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
68 views

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
votes
0answers
55 views

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 ...
8
votes
1answer
144 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
votes
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
183 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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votes
3answers
97 views

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 ...
0
votes
1answer
79 views

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
votes
1answer
122 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
votes
2answers
97 views

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
136 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 ...
2
votes
3answers
127 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 ...
5
votes
2answers
202 views

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 ...
16
votes
3answers
659 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
108 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
206 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
80 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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vote
2answers
147 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
votes
1answer
240 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}} + ...
4
votes
2answers
207 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 ...
4
votes
4answers
121 views

Why do $f$ and $f'$ generate all of $K[X]$?

I have been studying Marcus' Number Fields. I am stuck on a remark in Appendix 2, page 258. He says: A monic irreducible polynomial $f$ of degree n over $K$ (a subfield of $\mathbb{C}$) splits into n ...
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votes
1answer
127 views

How to remove intersection of ideal $I$ and $J$ from union of ideal $I$ and $J$

after get the intersection of ideal $I$ and ideal $J$ how to remove this intersection from union of ideal $I$ and ideal $J$ in order to do prime decomposition how can it do in maple? actually i ...
2
votes
2answers
205 views

Product of ideals corresponding to vanishing of points is equal to their intersection

Let $k$ be some field, and let $v,v',v''$ be three distinct points in $k^3$. Let $\mathfrak{m}_v = (X_1 - v_1,X_2 - v_2,X_3 - v_3)$ be the ideal in $k[X_1,X_2,X_3]$ corresponding to the polynomials ...
4
votes
3answers
296 views

Points and maximal ideals in polynomial rings

Let $k$ be a field, then I want to prove the following statement: for every $P=(b_1,\ldots,b_n)\in K^n$, the ideal $\mathfrak{m}_P=(x_1-b_1,\ldots,x_n-b_n)$ is maximal in the polynomial ring ...
3
votes
1answer
54 views

Radical Ideals: Show that $\sqrt{\sqrt{I}+\sqrt{J}}=\sqrt{I+J}$

The direction $\sqrt{\sqrt{I}+\sqrt{J}}\supset \sqrt{I+J}$ is trivial as $\sqrt{I}+\sqrt{J} \supset I+J$ since $\sqrt{K} \supset K$ for any ideal $K$. Is the following correct? My attempt: Suppose ...
3
votes
4answers
377 views

Understanding the quotient ring $\mathbb{R}[x]/(x^3)$.

I am having difficulty in understanding exactly the elements of the set $\mathbb{R}[x]/(x^3)$. I'll explain my thought process. The Quotient Ring is the set of additive cosets, so we have that ...
3
votes
1answer
164 views

Prime ideal decomposition in quadratic field extensions

Once you have the character $\chi$ of a quadratic field extension and the corresponding modulus $N$, it is easy to see which prime ideals split, ramify and are inert by looking at their remainder ...
9
votes
1answer
222 views

If $I$ is a finitely generated ideal of $A[X]$, is $I\cap A$ necessarily finitely generated for a commutative unital ring $A$?

Let $A$ be a commutative ring with $1$ and $A[X]$ the ring of polynomials in one variable over $A$. Assume $I$ is a finitely generated ideal of $A[X]$. My question is Is $I\cap A$ necessarily ...
1
vote
1answer
180 views

Finding finite basis of an ideal!

I have the following question... Set $I:= \{f \in\mathbb{Q}[X,Y] \mid \bar{f}(0,0)=0={\bar{f}(2,3)}\}$ I have proven that $I$ is an ideal... but the second part to the question is .. Find a finite ...
0
votes
1answer
31 views

Intesection of Ideals

Let P denote prime ideals,an $A_1,A_2$ ideals, i need to know how can i describe $\{P | A_1\subset P\}\cup\{P | A_2\subset P\}$ I mean it is equal to $\{P | A_1\cap A_2\subset P\}$ I think not, ...
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vote
0answers
69 views

Looking for a binomial system solver

I am interested in solving binomial systems of the form $$ \begin{cases} a_1 x_1^{d_{11}} x_2^{d_{12}} \cdots x_n^{d_{1n}} + b_1 x_1^{d_{11}} x_2^{d_{12}} \cdots x_n^{d_{1n}} &= 0 \\ ...
6
votes
3answers
180 views

Finding the ideals in a ring of fractions

I am dealing with the ring $$R=\left\{\frac{a}{b} \mid a,b\in\mathbb{Z}\mbox{, $b$ is not divisible by 3}\right\}$$ with addition and multiplication as defined in $\mathbb{Q}$ and I'm trying to find ...
1
vote
0answers
189 views

An Algorithm to Find the Generators of the Radical of a Monomial Ideal

Working over $R=\mathbb{C}[x_1,...,x_n]$, I'm given a ring homomorphism with $i\in{1,...,n}$ and $t\in \mathbb{C}$. $\phi_{i,t}(x_j)=x_j$ for $j\neq i$ to themselves. From this I've proven that an ...
7
votes
1answer
161 views

Prime elements in $\mathbb{Q}[[X,Y,Z]]$ whose status as an infinite series is unchanged by arbitrary multiplication

Let's suppose $R$ is the ring $\mathbb{Q}[[X,Y,Z]]$. I'm interested in finding power series $f(x,y,z) \in R \setminus \mathbb{Q}[X,Y,Z]$ which are, first of all, prime elements in $R$, but also ...