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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Finding associated primes of quotient modules

Consider the ideal $I=(a)\cap(a,b)^2=(a^2,ab)$ in $k[a,b]$ and set $R=k[a,b]/I$. The problem is to show that for all $n\ge 1$ and all $\lambda\ne 0$, the ideals $(b^n)$ and $(a+\lambda b^n)$ of $R$ ...
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Proof of every non-unit belongs to some maximal ideal. [duplicate]

I want to prove that every non-unit belongs to some maximal ideal. I did the following. Consider a commutative ring $R$ with unity. Consider a maximal ideal $M$. Also consider an element $r\notin M$. ...
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calculate intersection of two polynomial ideals

Show that in $\mathbb{C}[X,Y]$, the ideals $(X^3-X^2,X^2Y-X^2, XY-Y, Y^2-Y)$ and $(X^2,Y)\cap (X-1, Y-1)$ coincide. Is this a radical ideal? I can show that $(X^3-X^2,X^2Y-X^2, XY-X, Y^2-Y) \subseteq (...
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Equivalent definitions of primary ideals [duplicate]

Here are two definitions of a primary ideal. An ideal $I\subset A$ is primary if $I\ne A$ and $xy\in I\implies$ either $x\in I$ or $y^n\in I$ for some $n> 0$. An ideal $I\subset A$ is primary if $...
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Homogeneous elements of an ideal over a quotient ring

Let $k$ be a field. Consider the ideals $I_1=(x),I_2=(y),J=(x^2,y)$ of $R=k[x,y]/(xy,y^2)$. Show that the homogeneous elements of $J$ are contained in $I_1\cup I_2$ but that $J\not\subset I_1$ and $J\...
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Using the `initialIdeal` function in Macaulay2

My understanding is that there's an initialIdeal function in Macaulay2 for computing intial ideals with respect to a grading, specifically in the ...
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Ideals in Laurent polynomials over a field

Be $F$ a field and let $I$ be any ideal in $F[X,X^{-1}]$. For any $f = \sum_{n\in \mathbb{Z}}a_nX^n \in F[X,X^{-1}]$, define deg$^-(f) := \min\{n \in \mathbb{Z} \mid a_n \neq 0\}$. Consider the set $\...
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A typo in Eisenbud's Theorem 3.10?

Let $R$ be a Noetherian ring and $M$ a f.g. $R$-module. Let $M'$ be a proper submodule of $M$ and let $M'=\cap_{i=1}^n M_i$ be a primary decomposition with $M_i$ a $P_i$-primary submodule. Part (c) ...
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Finding inverses in quotient rings

In $ A=\mathbb{Z}[i]=\{a+bi \ : \ a,b \in \mathbb{Z}\} $ we consider $a=7+56i; \ b=3+3i; \ c=1+8i$. We will write $(a)$ to refer to the ideal generated by $a$ Find out whether the elements $\...
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Let $k$ be a non-algebraically closed field and $I\subset k[x_1,\dots, x_n]$ be maximal ideal. Is $V_{\bar{k}}(I)$ necessarily finite? [closed]

Let $k$ be a non-algebraically closed field and $I\subset k[x_1,\dots, x_n]$ be a maximal ideal. $\textbf{Q:}$ Is $V_{\bar{k}}(I)=\{x\in\bar{k}^n\vert \forall f\in I, f(x)=0\}$ necessarily finite?
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Product of ideals in $\mathbb{Z}[X]$

Consider the ideals $I = (2,X), J = (3,X) \in \mathbb{Z}[X]$. I want to show that the 'product set' $\Pi := \{ij \mid (i,j) \in I \times J\}$ is not an ideal in $\mathbb{Z}[X]$ and in particular, ...
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$\sigma$-ideal of subsets of $2^\omega$

I do not understand why here on the page 148 $\cal M^*_{2, K}$ is a $\sigma-$ideal of subsets of $2^\omega$. I even do not know the weaker statement: why it is an ideal?
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$R$ commutative ring with 1 and not every ideal is principal. Prove $R$ has ideal that is not principal.

I am wondering how to go about proving this, Let $R$ be a commutative ring with identity such that not every ideal of $R$ is principal. A) Use Zorn's lemma to show that $R$ has an ideal $J$ such ...
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Confused about quotient ring

For $I$ an ideal of a commutative ring $R$, I'm confused about the object $R/I$. My understanding is that this is defined by $R/I := \{rI: r \in R\} = \{\{ri: i \in I\}: r \in R\}$. However, by the ...
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Show that every non-zero ideal of the ring of rationals with odd denominator is generated by $2^{n}$ [duplicate]

Let $R \subset \mathbb{Q}$ be the subring $\left\{\frac{a}{b} \mid a, b \in \mathbb{Z}, b \text { odd }\right\}$. Prove that the ideals of $R$ are the zero ideal $\{0\}$ and $2^{n} R$ for $n \geq 0$. ...
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Proving $y-x^2, z-xy$ generate the ideal of the twisted cubic

This question comes from Hartshorne's exercise 1.2. He defines $Y:=\{(t,t^2,t^3)\in \mathbb{A}^3\mid t\in k\}$ and asks us, among other things, to find generators for the ideal $I(Y):=\{f\in k[x,y,z]\...
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Abstract Algebra: Quotient ring, ideal, and isomorphism

I need help with the following exam exercise, my teacher didn’t post the answer and I can’t manage to solve it. In $ A=\mathbb{Z}[i]=\{a+bi \ : \ a,b \in \mathbb{Z}\} $ we consider $a=7+56i; \ b=3+...
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Is this an example of comaximal ideals $I,J$ such that $IJ\not=I\cap J$?

Let $R$ denote the set of all finite formal sums of elements in the free group $\left<a,b\right>$ with the relation $a+b=1.$ Let $I$ be the principle ideal generated by $a$ and $J$ the principle ...
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$M_{i+1}/M_i\simeq R/P_i$ for some prime ideal $P_i$

Proposition 3.7 in Eisenbud says that for a f.g. module $M$ over a Noetherian ring there is a chain $$0=M_0\subset M_1\subset\dots\subset M_n=M$$ with $M_{i+1}/M_i\simeq R/P_i$ for some prime ideal $...
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Does a ring homomorphism $\phi: R \rightarrow S$ give rise to any map $\psi: R/I \rightarrow S/J?$

Let $R, S$ be rings. Suppose $\phi: R \rightarrow S$ is a ring homomorphism. Clearly we have a map $R \rightarrow S/J$ defined by the canonical map. However, for any ideal $I \subset R,$ can I ...
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There exists a prime ideal not containing non-nilpotent element [duplicate]

I've been trying to solve the following problem. Let $A$ be a commutative ring with identity and $a \in A$ a non-nilpotent element, i.e., $a^m \neq 0$ for all $m \in \mathbb{Z}^+$. Prove there exists ...
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Localization of the polynomial ring at a prime ideal modulo maximal ideal is isomorphic to polynomial ring modulo prime ideal.

Let $p \in K[T]$ irreducible, s.t. $\text{LC}(p) = 1$. Then $$ K[T]/(p) \cong K[T]_{(p)}/pK[T]_{(p)}.$$ What I have is: \begin{align*} &K[T] \hookrightarrow K[T]_{(p)} \text{ and } K[T]_{(p)} \...
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1answer
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If $R$ is an n-fir, why free $R$-modules of rank at most $n$ have unique rank?

I am studying the Cohn's book "Free ideal Rings and Localization in General Rings", and there is something he takes for granted and I cannot find its reason in any part of the book. The question is: ...
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Is $ \{ (\alpha ^ 2, \alpha) | \alpha \in \mathbb{R}\} \subseteq \mathbb{C}^2 $ an algebraic variety?

Is $S = \{ (\alpha ^ 2, \alpha) | \alpha \in \mathbb{R}\} \subseteq \mathbb{C}^2 $ an algebraic variety? I think the answer is no, and this was my approach: If $S$ is an algebraic variety then $I(S)$...
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How to find an ideal of a ring

I'm fairly new to ideals in my algebra course, and I understand the basics of ideals, such that I is an ideal iff (I,+) is a subgroup of (R,+) (using normal subgroup tests) and for all $r \in R$ and $...
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Find an element that generates the Ideal

Given the ideal: $A$:={$\sum_{i∈I}T^i|$ $I $ finite,$|I|$ even} ⊂ $\mathbb{F}_2[X]$ find an element that generates it. Now I know that it has to be $A=a$ $\mathbb{F}_2[X]$, where $a$ is the element ...
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Units in a discrete valuation ring.

I'm doing problem 2.26 in the book "algebraic curves" by Fulton: http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf One is given two DVR's R and S both of which have the same field of fractions $K$...
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Why is $6\mathbb{Z} + 9\mathbb{Z}$ a principal ideal of $\mathbb{Z}$?

Ler $R$ denote a commutative ring with identity. For $R = \mathbb{Z}$, why is the ideal $6\mathbb{Z} + 9\mathbb{Z}$ principal? Any help where to start would be appreciated.
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Finding all ideals $T$ in number ring $\mathbb Z[\sqrt{-3}]$ s.t. $\langle 4 \rangle \subset T$.

I want to find all ideals $T$ in number ring $\mathbb Z[\sqrt{-3}]$ s.t. $\langle 4 \rangle \subset T$. My idea: $ 4= 2\times2 = (1-\sqrt{-3}) (1+\sqrt{-3})$ so $\langle4\rangle = \langle 2\rangle \...
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Is this ideal principal?

Let $I = \{ f(x)∈ \mathbb{Z}[x] | f(0) \text{ is an even integer}\}$ This is an ideal of the ring $R = \mathbb{Z}[x]$. Is it principal? I have the definition of a principal ideal but I'm unsure if ...
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Product of the two subrings

These are problems bothering me. Let $S$ and $T$ are subrings of a ring $R$. And $I$ and $J$ are ideals of the $R$. Question 1 Let product of subrings like ideal product $IJ$ which means $ST = \{...
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How to prove this isomorphism of a quotient ring

I'm trying to understand more about ring theory and the concept of ideals has been confusing. I'm trying to understand why this is true: $\mathbb Z[\sqrt{-5}]/(1+\sqrt{-5})\simeq\mathbb Z/6\mathbb ...
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Ideal of $\mathbb{Z}[x]$ generated by $3$ and $x^2+1$ is proper

Let $I$ be the ideal of $\mathbb{Z}[x]$ generated by $3$ and $x^2+1$. Show that 1) $I$ is proper, 2) $\phi_a(I)=\mathbb{Z}$ for all $a\in\mathbb{Z}$, where $\phi_a(I):=\{f(a)\mid f\in I\}$ (...
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Ideal generated by two elements is maximal in $\mathbb{Z}[x]$

The question is to show that $I = (x^4 + x^3 + x^2 + x + 1, 2) \subset \mathbb{Z}[x]$ is a maximal ideal. I'm familiar with the results that $R / I$ is a field iff $I$ is maximal, and $R/I$ is a ...
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How to prove that $\dim_k(k[x_1,\dotsc,x_n]/\sqrt{I})=|V(I)|$?

Let $k$ be an algebraically closed field. How to prove that $\dim_k(k[x_1,\dotsc,x_n]/\sqrt{I})=|V(I)|$? I know that Hilbert Nullstellensatz will be required, but I can't figure out how. With the ...
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Find a homomorphism that maps a point in a Boolean algebra into the image of a proper filter

Let $\mathbb B$ be a Boolean algebra. Let $F$ be a proper filter in $\mathbb B$ (i.e. $0 \notin F$), and let $I$ be its dual ideal. Suppose there exists $a \in \mathbb B$ such that $a \notin F \cup I$....
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Does the vector space spanned by a Gröbner basis depend on the monomial order?

Let $k$ be a field, $n$ a natural number, and $I$ an ideal of $k[x_0,\dots,x_{n-1}]$. Given a monomial order $M$ on $k[x_0,\dots,x_{n-1}]$ let $G_M$ be the Gröbner basis of $I$ with respect to $M$ and ...
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On a colon ideal in the polynomial ring $\mathbb R[x,y]$

Consider the ring $R=\mathbb R[x,y]$. Let $\mathfrak m=(x,y)$. Let $n\ge 3$ be an odd integer and let $I_n=(x^n,y^n)$. What is the smallest integer $s\ge 1 $ (obviously depending on $n$) such that $(...
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Suppose $f,g\in \sqrt{I} \subseteq K[x]$ such that $in(f)>in(g)$. Then $g\in I$.

In reference to this answer. I'm guessing it has something to do with being able to find some elements in I with the same initial terms as f and g, and then canceling out leading terms somehow, but I ...
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How to determine the kernel $\ker \varepsilon_{\sqrt 2}$ and the image $\varepsilon_{\sqrt 2}(\mathbb Q[X])?$

Consider the $\mathbb Q$-algebra homomorphism $\varepsilon_{\sqrt 2}:\mathbb Q[X]\rightarrow \mathbb C$ defined by $\varepsilon(X)=\sqrt 2$. How to determine the kernel $\ker \varepsilon_{\sqrt 2}$ ...
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Is $(a)/(a^2)\cong R/(a)$, or something else? ; For ideals $(a), (a^2)\leq R$

(Rings are commutative with 1) In a lecture on introductory commutative algebra, I was presented with an exercise that basically just asked: For a non zero-divisor $a\in R$, we have $(a)/(a^2)\...
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1answer
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Polynomial Ring modulo Ideal is Polynomial ring of cosets of indeterminates.

I wonder if for any arbitrary ideal $I \leq K[X_1,\ldots,X_n]$, the following is true: $$ K[X_1,\ldots,X_n] \text{ mod } I = K[X_1 + I, \ldots, X_n +I].$$ If so, how can one show that?
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How to find the generator of the following ideals?

How to find the generator of the following ideals $\cal a=${$F\in \mathbb Q[X]:F(i)=0$} in $\mathbb Q[x]$, $\cal b=${$F\in \mathbb Q[X]:F(\sqrt 2i)=0$} in $\mathbb Q[x]$? $\cal c=${$F\...
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1answer
66 views

For an integral domain $R$, the rings $R\times R $ and $R\times R\times R$ are not isomorphic

For an integral domain $R$, the rings $R\times R $ and $R\times R\times R$ are not isomorphic My attempt: On contrary suppose that both are isomorphic then if G is prime ideal of one ring then its ...
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1answer
41 views

The localization of an ideal is equal to the localization of the ring

Suppose $m\subset R$ is a maximal ideal. Suppose $I\subset R$ is an ideal. I'm trying to understand these claims: If $m$ does not contain $I$, then $I_m=R_m$ as localizations of $R$-modules. If $m$ ...
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39 views

Is it true “If $X \subset Y$ then $\Bbb I(Y) \subset \Bbb I(X)$”(proper inclusion)?

Is it true "If $X \subset Y$ then $\Bbb I(Y) \subset \Bbb I(X)$; here I am using proper inclusion. Couldn't prove it though. Trying for long time please help. Actually I saw here "https://people.maths....
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coprime ideals in $K[X]$

If $K$ is a field, $A=K[X]$, take $m,n \in K$ such that $m \ne n$. Prove that the ideals $I=(X-m)$ and $J=(X-n)$ are coprime. I know the regular definition of coprime. But here, should we prove $I + ...
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On monomial ideals and ring generated by monomials

Question 1: Is $(x^4,x^3y,x^2y^2,xy^3,y^4)$ a maximal ideal in $\mathbb C [ x^4,x^3y,x^2y^2,xy^3,y^4] $? Question 2: Are the ideals $(x^4,x^3y,x^2y^2,xy^3,y^4)$ and $(x^4,x^3y,xy^3,y^4)$ distinct in ...
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1answer
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Restricting a filter in a Boolean algebra to a generating set and have it generate a filter

Let $B$ be a Boolean algebra and $S \subseteq B$ be a subset that generates $B$. Is it the case that every filter $x$ of $B$ is equal to the filter generated by $x \cap S$? What if $S$ itself is a ...
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When square of an ideal is the square of a maximal ideal in a polynomial ring

Question (1): let $J$ be an ideal in $\mathbb C[X,Y]$ such that $J^2=(X,Y)^2$, then is it true that $J=(X,Y)$ ? Question (2): let $J$ be an ideal in $\mathbb C[X,Y,Z]$ such that $J^2=(X,Y,Z)^2$, then ...