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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When are all (prime) ideals of an $R$-algebra, extensions of (prime) ideals of $R$?

Let $f:R\rightarrow R'$ be a homomorphism of commutative noetherian rings. When are all (prime) ideals of $R'$ extensions of (prime) ideals of $R$? Is it true for the case $R'$ is $R$-flat?
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principal ideals, integral domains, ideals,?

I am stuck trying to grasp this concept. I know that $\Bbb{Z}$ is a PID, $R=\Bbb{Z}[X]$ is not a PID, $\Bbb{Z}[i]$ is a PID. If someone could help me grasp these concepts it would be helpful. ...
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Finding a ring isomorphism

Let $\phi : R \to R'$ be a ring epimorphism and $J\lhd R'$ an ideal of $R'$. Indicate a ring isomorphism $\psi: R/\phi^{-1}(J) \to R'/J$ The only thing i know about this problem is that ...
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16 views

About the connection between ideals and homomorphisms

I know that for a homomorphism of rings $\psi : R\rightarrow S$ we have that $\ker\psi$ is an ideal of $R$. I was wondering if the opposite direction is true: Let $I$ be an ideal of $R$. Then does ...
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29 views

How to find the ideals of $\Bbb{Z}_n$

I have a homework problem to find the maximal ideals in $\Bbb{Z}_8$, $\Bbb{Z}_{10}$, $\Bbb{Z}_{12}$, and $\Bbb{Z}_n$. That question has already been asked on here, but I don't even understand how to ...
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40 views

Ideals generated for commutative ring

The following is a problem from the Gallian book. I'm trying to understand what exactly this ideal is and how to verify that it is in fact an ideal. "Let $R$ be a commutative ring with unity and let ...
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60 views

How can I prove an ideal is a product of two irreducible ones

I'm trying to solve this question: I have a guess that $(6+\sqrt{11})=(2,4+\sqrt{11})(2,-3\sqrt{11})$ using some formulas in this book page 48. However I couldn't verify if the multiplication of ...
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What are the open and closed sets in $\operatorname{Spec}\mathbb{C}[x,y]/(y^2-x^3)$?

What are the open and closed sets in $\operatorname{Spec}\mathbb{C}[x,y]/(y^2-x^3)$ ? $\operatorname{Spec}\mathbb{C}[x,y]/(y^2-x^3)=\{ (0),\ (\tilde{x}-a,\tilde{y}-b),\ b^2=a^3\}$.
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Factorisaing ideals in quadratic number fields

Show there is an ideal $a$ in $\mathbb{Z}[\sqrt-29]$ satisfying the equality $\langle8\rangle=a^{2}$. I tried to factorise the minimal polynomial over $\mathbb{F}_{8}$ but it does'nt seem to work, ...
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Dedekind's criterion clarification

Dedekind's criterion gives a way of factoring $p\mathcal{O}_K$ into prime ideals. (See http://math.stanford.edu/~conrad/154Page/handouts/dedekindcrit.pdf) Is it true that the prime ideals ...
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48 views

Symmetric powers of ideal quotients in a local ring.

Let $R$ be a local ring and $I \subset R$ any ideal. When is it the case that $(I \: \backslash I^2)^n = I^n \: \backslash I^{n+1}$? Put another way, when is the natural map $\text{Sym}^n(I/I^2) ...
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2-sided ideal in $\mathfrak{M}_{2\times 2}(\mathbb{Z})$ ring

Hi there: I know $\mathfrak{M}_{2\times 2}(\mathbb{Z})$ has left and right ideal, but is it true it does not have 2-sided ideal? If there is, could you give me an example. Thank you very much.
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Finding generators for an ideal of $\Bbb{Z}[x]$

We know that $\Bbb{Z}$ is Noetherian. Hence, we can conclude that $\Bbb{Z}[x]$ is Noetherian, too. Consider the ideal generated by $\langle 2x^2+2,3x^3+3,5x^5+5,…,px^p+p,…\rangle$ for all prime ...
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34 views

Principal ideals as generated groups

This seems like a pretty simplistic question, but I can't find a solid, non-ambiguous answer to it. The question I'm given: Is $I$ a principal ideal of $R$? Given: $R=\mathbb{Z}$ and $I=\left\langle ...
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42 views

How to decide if an ideal in $\mathbf Q[X,Y]/(P)$ is principal?

Let $P(X,Y)$ be an irreducible polynomial in $\mathbf Q[X,Y]$. Given an ideal $I$ of the quotient ring $\mathbf Q[X,Y]/(P)$ (say given by a set of generators) how can I decide if $I$ is principal or ...
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28 views

Let $f(x)\in F[x]$ ($F$ a field) be irreducible and let $\alpha$ be a root of $f(x)$. Then $h(x)\in(f(x))\Leftrightarrow h(\alpha)=0$?

Let $F$ be a field and $f(x)$ an irreducible polynomial in $F[x]$ such that $\alpha$ is a root of it: $f(\alpha)=0$. Now, let $(f(x))\subset F[x]$ denote the ideal generated by $f(x)$. My question is: ...
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39 views

Krull dimension, commutative algebra. Eisenbud, Exercise 10.3

This is the exercise. Let $k$ be a field. Prove that $k[x]\times k[x]$ contains a principal ideal of codimension $1$, although it's not a domain. Now, I have to find a principal ideal prime, such ...
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32 views

Factorisation in rings of algebraic integers

Determine the prime factorisation of the principal ideal $\langle15\rangle$ in $\mathbb{Z}[\sqrt{-14}]$. Can I use the fact $\langle15\rangle=\langle3\rangle\langle5\rangle$ and the factorisation ...
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1answer
42 views

Isomorphism between a sub-ring to $\Bbb Q$, the rational field

Let $R$ be a commutative ring with an identity, which contains $\Bbb Z$, the integer field, as a sub-ring with the same identity element. Let $I$ be a maximal ideal in $R$. I need to prove that if ...
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1answer
53 views

Addition and Multiplication table for Ring/Ideal

I'm not sure if it's possible to show it here, but how would the addition and multiplication table look like for R/I (where R is rings with ideal I) when $$ R = Z_{12} \text{ and } I = \{0,3,6,9\} ...
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Lattice with $3$ operations.

If $R$ is a commutative ring and $\mathcal I(R)$ denotes its set of ideals then I know that $\mathcal I(R)$ can be looked at as a complete lattice with intersection $I\cap J$ and addition $I+J$ as ...
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59 views

A theorem about ideals of $K[T_1,\ldots,T_n]$ and their generators

Suppose that $L\subseteq K$ is a field extension ( we are in characteristic $0$) and moreover that $\mathfrak a\subseteq K[T_1,\ldots,T_n]$ is an ideal ($T_1,\ldots,T_n$ are indeterminates). I have ...
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Product of two ideals in Dedekind domains

Let $\mathcal{O}$ be a Dedekind domain and $I=(x_1,\ldots, x_n),J=(y_1,\ldots, y_m)\subseteq \mathcal{O}$ two ideals. Is it possible, that $IJ\neq K$ with $K$ the ideal generated by the products ...
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76 views

Computing the radical of an ideal

What is the best way to compute $\sqrt{(X^2-YZ,X(1-Z))}$ ? This is after using Nullstellensatz by the way as I thought it would be easier to compute a radical than finding the vanishing ideal.
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27 views

A problem involving ideals and prime ideals. [duplicate]

Please help me with a solution to this problem. Let $R$ be a commutative ring. Let $A_1, A_2$ be two ideals of $R$, and $P_1, P_2$ two prime ideals of $R$. Assume that $A_1 \cap A_2 \subseteq P_1 ...
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2answers
56 views

How to show that $(Y- X^2, Z - X^3) \subseteq k[X,Y,Z]$ is a prime ideal?

I suppose that $k$ is an algebraically closed field (actually, my goal is to show $\mathcal{I}(\mathcal{V}(Y- X^2, Z - X^3)) = (Y- X^2, Z - X^3)$). (But I think algebraically closed is not necessary ...
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1answer
74 views

Prove that $m^2$ is primary

Let $m$ be a maximal ideal. I'm having a hard time proving that $m^2$ is primary. Let ${xy\in m^2}$ so $xy=t_{1}s_{1}+...+t_{n}s_{n}$ where the $t_{i},s_{i}$ are in $m$.
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When is nilradical not a prime ideal

Atiyah gives this criterion for nilradical to be a prime ideal.Nilradical is the intersection of prime ideals.Is nilradical prime iff there is only one prime ideal? ie Intersection of distinct prime ...
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1answer
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$I(V_1)+I(V_2) \neq I(V_1 \cap V_2)$?

Let $V_1,V_2 \subset \mathbb{A}^n(k)$ affine varieties ($k$ field). I've proved $I(V_1)+I(V_2) \subset I(V_1 \cap V_2)$, but I don't know how to prove $\supset$. I think that's maybe because that ...
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Number of elements in the ring $\mathbb Z [i]/\langle 2+2i\rangle$

The question is : Show that $I=\langle 2+2 i\rangle$ is not a prime ideal of $\mathbb Z[i]$. Also find the number of elements in $\mathbb Z[i]/I$ and its characteristic. My try: I started with ...
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Ideals - A Geometric Interpretation?

The standard way to define an ideal is as follows: $I$ is an ideal if it satisfies the following conditions: $(I,+)$ is a subgroup of $(R,+)$ $\forall x \in I$, $\forall r \in R :\quad ...
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1answer
47 views

$(2,1+\sqrt{-5})$ has integral basis $2$, $1+\sqrt{-5}$

$2,1+\sqrt{-5}$ is an integral basis for the ideal generated by them in $\mathbb{Z}[\sqrt{-5}]$. Is there a quick way to see this? What if these two are replaced with another pair? My method: Write ...
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1answer
68 views

Infinite sum of ideals

I've been trying to prove that given a ring $R$ and a collection of ideals $\{I_{\beta}\}_{\beta \in B}$ in $R$, the set $$\sum_{\beta \in B} I_{\beta}=\{f_1+\cdots+f_r:f_j \in I_{\beta_j}, \mbox{ for ...
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1answer
87 views

Minimal primary decomposition

Let $m$ be an integer ${\geq}3$ and $f(x,y,z)=y^m(x+y^3)-z^3$ in $k[x,y,z]$. Find the singular points of $f$ and find a minimal primary decomposition of the jacobian of $f$. I find the set of ...
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1answer
63 views

Equality with powers of an ideal

Let $A$ be an arbitrary (commutative with an identity) ring. Suppose $\alpha$ is an ideal. Is it true that $$\alpha(\alpha\cap\alpha^2\cap\alpha^3\cap…)=\alpha\cap\alpha^2\cap\alpha^3\cap…?$$ ...
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Examples of PIDs and prime ideals

(a) Give a specific example of a PID with exactly two prime ideals. Give a brief proof of your answer. (b) Give an specific example of a PID with infinitely many prime ideals. Give a brief proof of ...
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Descending chain condition on a finite dimension algebra

In a proof I'm reading, the author says "As $A$ is finite dimensional, a descending chain of left ideals must stabilize." The context is that $A$ is a finite dimensional simple $k$-algebra i.e. it ...
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56 views

Example of Localization and Prime Ideals

For each $n\in \mathbb Z^+$, give an example of a localization of $\mathbb Z$ with exactly $n$ prime ideals. Justify your answer. Could an example have something to do with a UFD or Noetherian ...
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Spectrum of a product of rings isomorphic to the product of the spectra

I've found in an exercise this statement: If $A$ is a commutative ring with unit and $A = A_{1} \times \dots \times A_{n}$ then $$\def\Spec{\operatorname{Spec}} \Spec(A) \cong \Spec(A_{1})\times ...
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Size of an ideal in a polynomial Ring

Let $F$ be a field and let $I = \{f(x) \in F[x]\mid f(a) = 0 ~~ \forall a \in F\}$. Prove that $I = \{0\}$ when $F$ is infinite. I have already shown that $I$ is an ideal and that $I$ is infinite ...
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1answer
51 views

Nilpotent Elements and Intersection of Prime Ideals

Prove that the set of nilpotent elements of a ring is the intersection of its prime ideals. I know these two useful facts: {nilpotent elements}$=\sqrt{0}$ $\sqrt{I}= \bigcap$ of prime ideals ...
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1answer
117 views

Ideals and prime ideals in a commutative ring. [closed]

Let $A_1$ and $A_2$ be two ideals, and $P_1$ and $P_2$ be two prime ideals in a commutative ring $R$. Assume that $A_1 ∩ A_2 ⊆ P_1 ∩ P_2$. Is there at least an $i$ and $j$ such that $A_i ⊆ P_j$ is ...
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1answer
42 views

Show an Ideal is the principal ideal for some polynomial.

Let $F$ be a field and $R = F[X]$. Suppose $I$ is an ideal of $R$. Show that $I = (p(X))$ for some $p(X)$ in $F[X]$. (Hint: consider a polynomial $p(X)$ of least degree in $I$.) I'm trying to do this ...
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1answer
61 views

generators of an ideal

I've been thinking about this exercise but I can't get the solution. In $\mathbb{R}^3$ , I consider the usual axis: $l_1=\{ x_1=x_2=0 \}$, $l_2=\{x_1=x_3=0\}$ and $l_3=\{ x_2=x_3=0 \}$. Calculate ...
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67 views

Problem about Gröbner basis.

I'd really appreciate if someone could help me. The problem is the following: If $\psi_1,...,\psi_m \in k[x_1,\dots,x_n]$ and consider the $k$-algebra homomorphism: ...
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1answer
49 views

Noetherian ring of Krull dimension $0$

I've found this claim: Let $A$ be a Noetherian ring of Krull dimension $0$ . Then $A$ is a field or it has a finite number of prime ideals. Why is this true ?
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prime ideals contains comaximal

Let $R$ be a commutative ring with unity 1 and $I$, $J$ and $P$ ideals in $R$ show that if every prime ideal of $R$ contains either $I$ or $J$ ,but not both then $I$ and $J$ are comaximal ...
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32 views

Proving the preimage of a ring homomorphism is not empty

I have a question from Abstract Algebra. Let $R$ and $S$ be commutative rings, and let $\phi:R \to S$ be a ring homomorphism. Show that if $J$ is an ideal of $S$, then $\phi^{-1}(J) := \lbrace r \in ...
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86 views

Vanishing polynomials

Let $K$ be a field and $V$ be the set of points $(t^3,t^4,t^5)$ where $t$ is in $K$. Set $I=(Y^2-XZ,Z^2-X^2Y,X^3-YZ)$. Show that $I$ is a subset of $A$, where $A$ is the set of polynomials which ...