Questions tagged [ideals]

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, ideals in Lie algebras.

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Minimal non-zero prime ideal in a UFD

I'm trying to understand the following Proposition: Proposition: Let $R$ be a UFD (unique factorization domain) with subset $P$ of the set of irreducible elements of $R$. Then $P$ is in bijective ...
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Construction of an ideal that is minimally generated by n elements

The following question is part of my abstract algebra assignment and I am looking for verification of solution. Question: Let A =k[x,y], where k is a field. For every $n\in \mathbb{N}$ , show that ...
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Corollary of Weyl's Theorem - direct sum of simple ideals

Weyl's Theorem states that if $L$ is a semisimple Lie algebra over $\mathbb{C}$, then any finite dimensional representation of $L$ is completely reducible. I want to show that a corollary of this is ...
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The local ring at the origin of $ \Bbb{A}^2 $

Consider the curves $ F = y-x^3 $ and $ G = y^3-x^4 $ over $ K. $ Find a polynomial representative of $ \frac{1}{1+x} $ in $ \mathscr{O}_0/ \langle F,G \rangle. $ I am having trouble simplifying $ \...
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Prove for the ideals $I,J$ that $J\nsubseteq I$

consider the ideals $I=\langle f_1,f_2\rangle$ and $J=\langle h_1,h_2\rangle$ in $\mathbb{Q}[x,y]$. I want to prove that $J\nsubseteq I$. I'm trying to do this by proving that every element from $J$ ...
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Is $S^{-1}(q_i)$ is a primary ideal in $S^{-1} A$?

This question was left as an exercise in class of commutative algebra and I am struck on it. Let A be a noetherian ring and $I=\cap_{i=1}^n q_i$ be a minimal primary decomposition. Let $p_i = \sqrt{...
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$lAnn_R (I)$ intersects nontrivially any non-zero right ideal of $R$

Let $I$ be a nilpotent ideal of ring $R$ (with unit). Then $lAnn_R (I)$ has a nontrivial intersection with any non-zero right ideal of $R$,where $$lAnn_R(I)=\{ r\in R\; |\; rI=0\}.$$ I was wondering ...
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$ \mathfrak mR_{\mathfrak m} $-primary ideal is the localization of some $\mathfrak m$-primary ideal?

Let $\mathfrak m$ be a maximal ideal of a commutative Noetherian ring $R$. Let $J$ be an ideal of $R_{\mathfrak m}$ such that $\mathfrak m^n R_{\mathfrak m}\subseteq J \subseteq \mathfrak mR_{\...
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Constructing semisimple Lie algebras of dimension $n$

Suppose we want to construct a semisimple Lie algebra $L$ of dimension $n$. We know that if we can write $L = L_1 \oplus L_2 \oplus \cdots \oplus L_s$ as a direct sum of simple ideals, then $L$ must ...
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Prime ideal of $\mathbb{C}[x,y]$.

I want to show that $J:= \langle x-y^2+1 \rangle$ is a prime but not a maximal ideal of $\mathbb{C}[x,y]$. My idea is that $x-y^2+1=x-(y-1)(y+1)$ and so $\langle x-y^2+1 \rangle \subsetneq \langle x, ...
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Prove that $Z(A)= \cup_{i=1}^n p_i$

I am reading commutative algebra from a class notes and I am not able to understand this proof. Statement: Let $p_i$ are primary ideals associated to I (or A/I). Then show that $Z(A)= \cup_{i=1}^n ...
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A question in 1st uniqueness theorem of primary decomposition

I am self studying commutative algebra from a class notes based on atiyah and macdonald and I am struck on this proof. Statement; Let $I=\cap_{i=1}^n q_i$ be a minimal primary decomposition of I. ...
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How to determine if the ideal $I = \langle x-1, y \rangle$ is a maximal ideal of $\mathbb{Q}[x, y]$

How do I determine whether $\mathbb{Q}[x, y] / I$ is a field? Where I is generated by the Gröbner basis $$ I = \langle x-1, y\rangle = \bigl\{ a(x,y)\,(x-1) + b(x,y)\,y \mid a,b \in \mathbb{Q}[x,y] \...
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A question regarding primary decomposition of ideals

This question was left as an assignment in my class of commutative algebra but I was not able to completely solve it. Prove that $I = (x^2, xy)= (x) \cap (x^2,y) = (x) \cap (x^2,xy,y^n)$ for any $n \...
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Is $\bigcap_{n=1}^\infty I^n$ contained in a minimal prime ideal?

Let $I$ be a proper ideal of a commutative Noetherian ring $R$. Let $J:=\bigcap_{n=1}^\infty I^n$. Then, is it true that $J \subseteq P$ for some minimal prime $P$ of $R$? By prime avoidance, I am ...
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Definition of uniform $ω_1$-dense ideal

Can one define the term uniform $ω_1$-dense ideal without the use of Boolean algebras?
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How can I finish the proof for the following proposition: every ideal in k[x] is a principal ideal? [duplicate]

Proposition : Every ideal in k[x] (polynomial ring) is a principal ideal Proof : Suppose that $I\subseteq K[x]$. Take $p(x)\in I$ such that $p(x)$ is monic and $deg(p(x))$ is minimal over all ...
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If $A$ is a ring, why isn’t the ideal $A^{(\mathbb{N})}\subset A^\mathbb{N}$ finitely generated?

I’m using the definitions: $\underline{a}=(a_i)_{i\in I} \in A^I $ (where $A$ is a ring), $$A^{(I)}:=\{ \underline{a} \in A^I \mid |\mathrm{supp} (\underline{a})| < \infty \} \subset A^I,$$ where ...
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$I=(2,x)$ and $J=(3,x)$ , a product of ideals

$I=(2,x)$ and $J=(3,x)$ be ideals. $I\cdot J=(6,x)$. Because $6=2\cdot 3$ and $x=3\cdot x-2\cdot x$, does this means that $I\cdot J$ contains $(6,x)$ or is it the inverse?
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Unit group of a quotient ring $(R/I)$

I have to show (or find a counterexample for) the following relation: Let $R$ be a ring, $I$ an ideal of $R$ and $R/I$ the quotient ring. If $a \in R^\times \Rightarrow a + I \in (R/I)^\times$ I have ...
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Given the ring $R:=(\mathbb{Z}/n\mathbb{Z})$ find its maximal ideals $M$ and the number of elements in $R/M$ [duplicate]

I recently stumbled across this question for the rings $R:=(\mathbb{Z}/8\mathbb{Z})$ and $R:=(\mathbb{Z}/30\mathbb{Z})$ and I was wondering wether this could be generalized for any given ring $R:=(\...
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Ideals of $C^0([0, 1]; \Bbb R)$ and compactness [duplicate]

Let $C := C^0([0, 1]; \Bbb R)$ the ring of continuous real functions on $[0, 1]$. Let $I \subset C$ an ideal. We suppose that $I$ is not contained in any $I_x:= \{f \in C \lvert f(x) = 0\}$. Show that ...
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Giving the size of the quotient subring $(\mathbb{Z}/30\mathbb{Z})/I$ where I is a maximal ideal [duplicate]

I need help with the proof of a specific step in the following problem: Given the ring $\mathbb{Z}/30\mathbb{Z}$ give the size of the quotient subring $(\mathbb{Z}/30\mathbb{Z})/I$ where I is a ...
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Two sided ideals of $k\left<x, y\right>$

Question: Are the two sided ideals of $k\left<x,y\right>$ (polynomial ring in twonon commuting variables) finitely generated (as two sided ideals) when $k$ is a field? I know that there are one ...
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Image of a bilinear form

Say that a commutative ring with one $A$ has Property (P) if the image $\beta(M\times N)\subset A$ of any bilinear form $\beta:M\times N\to A$ is an ideal of $A$. Here $M$ and $N$ are two $A$-modules. ...
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principal ideal and ideal generated by subset S

Let R be a commutative ring with unity and a in R. Then $<a>$=aR=Ra We can easily show that aR=Ra, aR ⊆ $<a>$ and Ra ⊆ $<a>$ Next I am trying to show that $<a>$ ⊆ aR We khow ...
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Definition of fractional ideal

I have a little problem with the definition of a fractional ideal. The definition I've been given is a set $f\subseteq Q=\text{Frac}(R)$ such that $\exists b\in R\backslash \{0\}$ such that $b.f\...
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Are rings $\mathbb {R}[X]/(X^2 + 1)$ and $\mathbb {R}[X]/(X^2 − 1)$ isomorphic?

So my proof is $\mathbb {R}[x]$ is a PID so it is an integral domain $(x^2 + 1)$ is irreducible as it has no roots and is of degree 2 since $\mathbb {R}[x]$ is an integral domain $(x^2 + 1)$ is prime ...
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Ring and generator ideals [duplicate]

Let $R$ be a ring and $r,s∈R$ So this is true If $r=st$ for some $t∈R^x$, then $(r)=(s).$ $R^x$ being the units in $R$ Why is it true and why is the converse not? If $(r)=(s)$, then $r=st$ for some $t∈...
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Is the Jacobson radical of a ring satisfying the descending chain condition on its two-sided ideals nilpotent?

Let $R$ be a ring with identity that satisfies the descending chain condition on its two-sided ideals. Is it true that the Jacobson radical $\text{Jac}(R)$ of $R$ is nilpotent? Suppose $R$ is ...
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For any ideal, why must a ring homomorphism be surjective for the ideal to be mapped to an ideal?

Problem: Let $f$: $R \to S$ be a ring homomorphism Show that if $f$ is surjective, then for any ideal $I \subset R$, the set $f(I)$ is an ideal of $S$. My Solution: Let $x,y\in I\subset R\implies f(x),...
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Is the ideal product presheaf a sheaf?

Given a ringed space $(X,\mathcal{O}_X)$ and ideal sheaves $\mathcal{I},\mathcal{J}\subset\mathcal{O}_X$, we define the ideal product presheaf $\mathcal{I}\cdot_p\mathcal{J}$ as the ideal presheaf $$ ...
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Let $I,J$ be ring then show that $V(I+J)=V(I) \cap V(J)$

$V(I+J)=V(I)\cap V(J)$ ,, ( V states zero set) I showed one way but the way that I ask you now is a problem for me. $V(I+J)=\{ P\in A^n: (F+G)(P)=0, \forall F+G \in I+J \}$ $={P\in A^n:F(P)+G(P)=0,∀F+...
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Given $K/F$ where $K$ is a field extension of $F$ of the form $F[x]/\langle p(x) \rangle$, what is the structure of $K[x]$?

$F$ is a field. $\langle p(x) \rangle$ is a maximal ideal. So $K = F[x]/\langle p(x) \rangle$ is a field extension. I am trying to understand what would be the structure of $K[x]$? $F[x]$ has all ...
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Problem with question 3.19 of Eisenbud (Commutative Algebra)

I'm with problem on understanding the suggestion that Eisenbud gives on one of his exercises. Suppose $R$ is a ring containing a field $k$, and let $I_1,\dots,I_n$ be ideals of $R$. If $(f_1,\dots,...
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Monomial in monomial ideal

Reading first time about monomial ideals and I am stuck with one of the very first results: Let $\mathbb{K} $ be a field and $\Lambda \subseteq \mathbb{Z}_{\ge 0}^n$. Given a monomial ideal $$  I= \...
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Extensions by Adjoining elements and Extensions by quotient of a Principal Ideal

Extensions can be constructed 2 ways to get an extension with roots of a polynomial Adjoining an element to a field - i.e. $F(\sqrt 2)$ is an extension of $F$. You can also build a tower of ...
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Prove/Disprove that ideal $\langle xy-z^7\rangle$ and $\langle xy,yz,zx\rangle$ are primary

I need to check whether the ideals $\langle xy-z^7\rangle$ and $\langle xy,yz,zx\rangle$ in the ring $K[x,y,z]$ primary or not ($K$ is a field) The only tool I know at the moment is that an Ideal is ...
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Maximal regular sequence coincides with system of parameters

I'm needing help in this question. Let $k$ be a field. Consider the $k$-algebra $R:=k[x,y,z,w]/(z+w,xy+xw)$ and define the ring $A$ the localization of $R$ in its maximal ideal $\mathfrak{m} = (\...
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How to approach proof of finding number of prime ideals lying over another prime ideal

Consider the following problem: Problem: Let $f= f(X) \in \mathbb{Z}[X]$ be a monic polynomial of positive degree and let $B=\mathbb{Z} [X] /\left<f\right>$. For a prime number $p$, show that ...
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Can the decomposition of a principal ideal in a Dedekind domain contain a non-principal ideal as a factor?

If $I= \prod_{P_i\in Spec(R)}P_i$ is the decomposition into prime ideals of a principal ideal $I$ in a Dedekind domain $R$, can one of the $P_i$ be a non-principal ideal? I guess it can’t, but I don’t ...
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Multiplicity of intersection in terms of degree of the quotient of ideal by its saturation

In this guide to Macaulay2 I found an interesting way to compute intersection multiplicity. On page 61 the authors give an explicit case of the following argument: We want to compute the intersection ...
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How to Prove an Ideal Can Be Generated From 2 Elements [duplicate]

Given a commutative ring with unity, $R$, an $a,b \in R$, and an ideal $I$ such that $I=\{ax+by \mid x,y \in R\}$. Prove that $I=(a,b)$. I think I want to show that $R$ is a PID, but I am not quite ...
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1 answer
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Localization an ideal by maximal ideal [duplicate]

I try to show this, suppose $R$ be a commutative ring with $1$ and $A$ be an ideal in $R$. Show if $A_{M}=0$ for all maximal ideal $M$, then $A=0$. My idea this, If $a\in A$ then for every maximal ...
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If factor ring is domain ( non commutative ), then quotient must be prime

Let $\frac{R}{P}$ be a domain ( not necessarily commutative ). Show that $P$ is prime. In previous questions asked on this site, integral domain ( commutative ) is assumed, but I don't think this is ...
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1 vote
2 answers
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Nilpotent ideals of algebra over a field.

I have 2 questions about the nilpotent ideals of an algebra $A$ over a field $K$. Each non-nilpotent left ideal of $A$ contains a nonzero idempotent element of $A$. The sum of all left nilpotent ...
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1 vote
1 answer
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Let $k$ be a field. Can all the ideals in $k[X,Y]$ be generated by at most two elements? [duplicate]

Let $k$ be a field. Can all the ideals in $k[X,Y]$ be generated by at most two elements? I don't think it can. I thought of the ideal $(X^2,XY,Y^2)$ and I think it can't be generated by only two ...
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1 vote
1 answer
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Prime ideal $\implies$ maximal in a Boolean ring

I want to show that a prime ideal in a non-unital Boolean ring $B$ is maximal ideal. If the ring contains unity then it is easy. As Boolean rings are commutative, for a prime ideal $P$ the ring $B/P$ ...
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2 answers
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Describe the ring structure quotient ring $Z[x,y]/(x^2,y^2,2)$.

This is taken from an exercise problem in Dummit and Foote's: Describe briefly the ring structure of $\mathbb Z[x,y]/(x^2,y^2,2)$ and show that $\alpha^2=1$ or $0$ for every $\alpha$ in the ring. I ...
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Example of ideal intersect subring that is non-ideal [closed]

Let $R$ be a ring with identity. If subrings are assumed to contain identity, does there exist an ideal $I$ and a subring $S$ of $R$, where $I\cap S$ is not an ideal of $R$?
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