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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disconnected set in Zariski topology

Suppose that $V$ is a nonempty affine algebraic set. If $k[V]$ is the direct sum of two non-zero ideals, prove that V is disconnected in the Zariski topology. I am preparing for the midterm, and I ...
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Prove that $\sqrt{I}=I$

Let $f(x,y)=y^2-x^2-x^3$ and consider the ideal $I=(f)$ in $\mathbb C[x,y]$. I want to show that the radical ideal $\sqrt{I}$ is equal to $I$. Any help? I've tried an approach using the division ...
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Questions about the quotient ring $(\mathbb{Z}/2\mathbb{Z})[x]/\langle x^2+x+1\rangle$

I'm just starting to learn about quotient rings. I was able to think about what type of of elements are generated by some $a$ when $\langle a \rangle$ is simply an integer, and also with simple ...
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If $A,B$ are ideals in $R$, show that $AB=\{\sum a_ib_i \colon a_i \in A, b_i \in B, n \in \mathbb{Z}^+\}$ is an ideal in $R$.

If $A,B$ are ideals in $R$, show that $AB=\{\displaystyle\sum a_i b_i \colon a_i \in A, b_i \in B, n \in \mathbb{Z}^+\}$ is an ideal in $R$. I am having trouble justifying that for all $s \in AB$ ...
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set of polynomials whose coefficient sum is zero is an ideal.

Show that the set of polynomials whose coefficient sum is zero is an ideal. Proof: Let $ p(x) = a_nx^n + ... +ax + a_0$ and $q(x) = b_ny^n + ...+ by + b_0$. Then suppose $a_n + ...+ a_0 = 0$ and ...
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Prove the following set is an ideal of the ring $Z[x]$

Show that the set of all polynomials whose constant term is a multiple of 3 is an ideal of the ring $Z[x]$. Attempt: Suppose then we need to show if $p(x), q(x) \in I$ then $p(x) + q(x) \in I$ ...
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Is some One Dimensional subalgebra an Ideal of the 2 Dimensional Non-Abelian Lie Algebra?

Is there any one dimensional subalgebra which is an Ideal of the two dimensional non-abelian Lie Algebra? i.e. is it invariant as a subalgebra of the 2D non-abelian algebra I read that "all the ...
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Wrong answer? Irreducible components of $Y$ defined by $x^2 - yz$ and $xz - x$

From Hartshorne's Algebraic Geometry, exercise 1.3: Let $Y$ be the algebraic set in $\mathbb A^3$ defined by the two polynomials $x^2 - yz$ and $xz - x$. Show that $Y$ is a union of three ...
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Is there any examples of a Banach algebra which every ideal of it, is a maximal ideal?

Is there any examples of a Banach algebra which every ideal of it, is maximal ideal? Or, Is there any conditions which turn all of the ideals of a Banach algebra to maximal ideals?
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Proving an ideal of polynomial ring

I posted this question here intended to include it as part of my writing. However, I got stuck again by something of entirely different nature: How do you prove that $(x_1), (x_1, x_2), (x_1, x_2, ...
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Why $1\in I\implies I=R$?

Written in Abstract Algebra by T. W. Judson : Theorem Let R be a ring with identity and suppose that I is an ideal in R such that 1 is in I. Since for any r∈R, r1=r∈I by the definition of an ...
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Neat way to find the kernel of a ring homomorphism

I'm trying to solve an exercise in Commutative Algebra in which I need the kernel of the following homomorphism: Consider the homomorphism $ \phi :k[[x,y,z]] \to k[[t]]$ which sends $x \to t^3,y ...
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Are $(X−λ),(X^3),(λ),(X^2−1)$ prime and maximal ideal in K[X]? [closed]

(i) $(X−λ),(X^3),(λ),(X^2−1)$ in K[X], where K is a field and and λ∈K∖{0}λ∈K∖{0}. (ii) $(13);(2,X^3+X^2+X+1)$in $\mathbb{Z}[X]$. ($(2,X^3+X^2+X+1)$ is an ideal generated by 2 and $X^3+X^2+X+1$ ...
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Why $IJ\subset I\cap J$ (for $I$ and $J$ ideal) whereas if $N$ and $H$ are groups $N,H\leq NH$

Let $N,H$ two subgroup of a group $G$ such that at least one is normal. By Surb answer here, $NH$ is the smallest group that contain $N$ and $H$. But if $I$ and $J$ are ideal, they are also group for ...
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How do we find a primary decomposition of an ideal?

Currently I'm reading about primary decomposition of ideals from Atiyah and Macdonald's Introduction to Commutative Algebra book. I've read all the theorems related to primary decomposition given in ...
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The ring $R/I$ is a total quotient ring iff $I$ is… what?

Conventions. All my rings are commutative. By a total quotient ring (TQR) , I mean a ring whose every regular element is a unit. Now let $R$ denote a ring and $I$ denote an ideal of $R$. The ...
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Ideals which contains an element

Let $\theta=\dfrac{1+\sqrt{-31}}{2}$, determine which ideals of $D=\mathbb{Z}[\theta]$ contains $1+\theta$. I know that if i.e $6\in\mathfrak{a}\Rightarrow \mathfrak{a}\mid 6D$ and then ...
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minimal generating set for modules over quotient ring of polynomial ring

Let $R=\frac{k[X_1,\ldots,X_n]}{I}$, where $k$ is a field and $I$ is an ideal. Let $M$ be a finitely generated module over $R$. I would like to compute a minimal generating set for $M$. As $R$ is not ...
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37 views

Appending field polynomials to an ideal produces a variety that excludes all elements from the algebraic closure of $k$.

In the paper "Algebraic Attacks on the Courtois Toy Cipher" written by M. Albrecht, he defined field polynomials and stated a corollary as follows: Definition: Let $k$ be a field with order ...
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In $\mathbb{Z}$ every subring is an ideal.

Prove that in $\mathbb{Z}$ every subring is an ideal. Proof: Let $S$ be a subring of $\mathbb{Z}$. Since $S$ is a ring, $(S,+)$ is a group. If $m\in \mathbb{Z}, s\in S$, then adding $s$ by $m$ ...
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Infinitesimal neighborhoods and ideal containment

Suppose I have the containment $I \subseteq P$ in some commutative ring $R$ where $P$ is a prime and $I$ is an ideal. Let $\sqrt{I} = P$. I am wondering if it is true that I can find some $n$ such ...
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41 views

Ideal generated by polynomials

Suppose $a$, $b$ and $c$ are distinct real numbers. Let $f_1 = (x - a)(x - b) $, $f_2 = (x - b)(x - c)$ and $ f_3 = (x - a)(x - c) $. Then $\langle f_1, f_2, f_3\rangle = 1$. How to approach this ...
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Solution of an equation in quotient-group?

$f(x)=2x^{3}+4x^{2}+2x+2$ and $g(x)=2x^{2}+x+3$ in $\mathbb{Z}_5[x]$ $$I=<f(x)>$$ I want to know if there is such natutal number $n$, and if there is to find, otherwise to prove that there ...
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Let $R$ be a ring and $I$ a subgroup under addition, show $I$ is an ideal

Let $R$ be a ring and $I$ be a subgroup under addition. Prove if for every $a,b \in R$ 1) $(a + I) + (b + I) = (a + b ) + I$ and 2)$(a + I)(b + I) = ab + I$ then $I$ is an ideal. attempt proof: ...
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Density of integers that are norms of ideals for $K \ne \mathbb{Q}$

I am interested in proving and understanding the following statement: If $K \ne \mathbb{Q}$, then the set of positive integers that are norms of ideals in $\mathcal{O}_K$ have density zero in ...
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Product of ideal generators

Can we in general say that if we have an ideal $(I,J)$ that this is the same as the ideal $(I,J,IJ)$, where $IJ$ is the product of I and J?
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Ideals of $\mathbb{Z}[i]$ geometrically

It is pretty easy to visualize the ideals of $\mathbb{Z}$ in the "integer line". Let's go up to $\mathbb{Z}[i]$ and consider the ideal $3\cdot\mathbb{Z}[i]$. We can visualize it as a "sub-lattice" ...
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Ideals in Lie algebras

Is it true that in a Lie algebra $\mathcal {L}$ the product of two ideals $[I, J]$ is equal to the intersection $ I\cap J $?
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A principal maximal ideal

Let $(R,m)$ be a local integral domain, and $t\in m^{-1}$ be such that $tm=R$. Is it true that $m$ is principal? If $t=a/b$ with $a,b\in R$ and $b\not =0$ then $ac/b=1$ for some $c\in m$, and ...
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What is a cubic ideal/partial cubic ideal?

Can anyone explain to me as simple as possible what a cubic ideal/partial cubic ideal is? I know what an ideal is in Ringtheory but I couldn't find anything about a cubic ideal or about partial cubic ...
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Prime ideals $\mathfrak{p} \supset \mathfrak{a}$ are finite in one-dimensional Noetherian domain

Let $A$ be a one-dimensional Noetherian domain. Let $\mathfrak{a} \neq 0$ be an ideal of $A$. How do I prove that prime ideals $\mathfrak{p} \supset \mathfrak{a}$ are finite? Thanks.
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$x=(0,\overline{1})$ and $y=(0,\overline{2})$ generate the same ideal in $R=\mathbb{Z}\times\mathbb{Z}/5\mathbb{Z}$

How do I show that $x=(0,\overline{1})$ and $y=(0,\overline{2})$ generate the same ideal in $R=\mathbb{Z}\times\mathbb{Z}/5\mathbb{Z}$, but that there is no $u\in R^*$ such that $y=ux$? Working with ...
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Property of the norm of an ideal

In Stewart/Tall's book on ANT they authors claim that if $ 0 \ne \mathfrak{a}$ is an ideal of $\mathfrak{O}_K$ the ring of integers of a number field $K$, then $N(\mathfrak{a}) \in \mathfrak{a}$. The ...
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Zero dimensional local complete intersection in $\mathbb{A}^2$

Denote by $\mathcal{O}_{p}$ the localization of $\mathbb{C}[x, y]$ at the ideal $(x, y)$. Let $I$ be an ideal of $\mathcal{O}_{p}$ generated by two elements. Moreover require that $\mathcal{O}_{p} / ...
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$\mathfrak{a\subseteq b}$ and $\mathfrak{bc}=\lambda D$ then there is an ideal $\mathfrak{b}'$ such $\mathfrak{a=bb'}$

Let be $D$ a commutative domain, $\mathfrak{a,b,c}\subseteq D$ ideals. Show that: if $\mathfrak{a\subseteq b}$ and $\mathfrak{bc}=\lambda D$ then there is an ideal $\mathfrak{b}'$ such ...
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If $A\subset \lambda D$ then $\exists A'$ such $A=\lambda A'$

Let be $D$ a commutative domain, $\lambda\in D\setminus \{0\}$ and $A\subset D$ an ideal. Then if $A\subset \lambda D$ then exists $A'\subset D$ such $A=\lambda A'$ my work: If $A=\lambda D$ ...
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Valid proof of Nakayama's Lemma?

Is this a valid proof of Nakayama's Lemma? I ask because I find the inductive step much more intuitive than the one I read in most authors... Proposition (Nakayama): Suppose that $R$ is a commutative ...
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Calculating a free resolution of $\mathbb Q[x,y,z]/I$ where $I = (x,y,z)$

Let $R=\mathbb Q[x,y,z]$ and $I = (x,y,z)$. I am trying to find the minimal free resolution of $R/I$. This is what I have got: $R \rightarrow R/I$ whose kernel is $I$, which is generated by 3 ...
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Show that $\operatorname{rad}(I)=\bigcap P$ for all $P$ prime containing $I$

Let $R$ be a commutative ring with identity and let $I$ be an ideal of $R$. Define $\operatorname{Rad}(I)=\{a\in R:\exists n\in\mathbb N, a^n\in I\}$. Show that $Rad(I)$ is the intersection of all ...
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A minimal primary decomposition of a radical ideal is a prime decomposition.

I want to prove that, if $I$ is a radical ideal in a Noetherian ring, and if $I=Q_1\cap\cdots\cap Q_r$ is a minimal primary decomposition (i.e., each $Q_i$ has a distinct radical, and no $Q_i$ ...
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Sanity check: prime ideals in number fields

Work in $\mathbb{Q}(\zeta_m)/\mathbb{Q}$. If $P$ is a prime ideal of norm $N(P)=p \hspace{1mm}\not \vert m$, does it follow that $p \equiv 1 \text{ mod }m$? I am sure this is not true, but am bad at ...
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An example of a variety of an ideal of a variety…?

I am asked to show an example of some $S \subset k^n$ such that $V(I(S)) \ne S$. So basically, $S$ is some set of points in $k^n$ a field of $n$ dimensions, and an affine variety, I guess. I've ...
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Ideal generated by polynomials and linear dependence

I've been thinking about this for almost a day and I have given up. I just get stuck in an invalid argument and dunno how else to do this. So the question is: Let $s>1$ and let $f_1,...,f_s$ ...
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Sub-group of the modulo group is an ideal

how do I show that every sub-group of ring $Z_n$ is an ideal in $Z_n$? If $n$ is prime, the only sub-groups are the trivial and that mean's they are ideals, but if $n$ isn't prime, there are non ...
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To factorize the given expression to get the following: $7+\sqrt{-5}=z+z \sqrt{-5}=z(1+ \sqrt{-5})$, where $z \in \mathbb{Z}$

Please, check my version: As $z \in \mathbb{Z} \Rightarrow z = a + ib$, where $a, b \in \mathbb{R}$ Thus $z(1 + \sqrt{-5}) = (a + ib)(1 + i \sqrt{5}) = a + ai \sqrt{5} + ib + {i}^{2} \cdot b ...
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how to do isomorphic ideal from root system of degree 5 or more [closed]

update1 i notice solving 1+x+x^2+x^3+x^4+x^5 have 5 solutions two conjugate real number and each of them having conjugate complex number part i change a*b + c to a1*a2*b1*b2 + c1*c2** still can ...
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2answers
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A example of a commutative chain ring [duplicate]

We say a commutative ring $R$ is a chain ring whenever its ideals form a chain with respect to inclusion. I am looking for a chain ring with Krull dimension two? Thank you for any help.
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Find the order of the quotient ring $\mathbb Z[x]/J$

Let $J=\{f(x)\in \mathbb Z[x]:6\mid f(0)\}$. Show that $J$ is an ideal of $\mathbb Z[x]$, but not a prime ideal of $\mathbb Z[x]$. Also find the order of the quotient ring $\mathbb Z[x]/J$. I ...
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Relations among monomials

Consider the polynomial ring $k[x_1,\ldots,x_n]$ for some field $k$. For each $\bar{a}=(a_1,\ldots,a_n)\in\mathbb{Z}^{\ge 0}$ let $$X_{\bar{a}}=\prod_{i=1}^nx_i^{a_i}$$ Let ...
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$\mathbb{Z}[i]$ is principal. And what are the units

I have elements of the form $a+bi$. I have attempted to consider arbitrary ideals in $\mathbb{Z}[i]$. If $N$ is ideal and $N=\{0\}$ then it is generated by $0$. If $N$ is not trivial, then exists ...