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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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 $6D=Q^2PP$ ...
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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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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 $q=p^n$...
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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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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 isn'...
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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 since $...
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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 $\mathfrak{...
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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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63 views

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 been ...
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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 \sqrt{...
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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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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 know ...
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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 $I=\{\bar{a_1},\ldots,\...
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$\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 $f=...
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Is the Jacobson radical of a ring with finite spectrum and nilpotent nilradical nilpotent?

I tried to solve 1.3.3 in Bosch, Algebraic Geometry and Commutative Algebra. I did not find a way to solve it. But I found this: Finitely many prime ideals ⇒ cartesian product of local rings. And ...
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Maximal ideals of polynomial ring $R[x]$ over a commutative ring $R$

In the article 1, Page 46, the author notes that: At this point it should be noted that there is an obvious connection with commutative rings. In fact, if $R$ is a commutative domain and $T= C$ ...
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Non-invertible elements form an ideal

Problem is this: Suppose for any element $r$ in a ring $R$ with unity $1$, $r$ or $1-r$ is invertible. Then show that non-invertible elements form an ideal of $R$. It is crystal clear that $-r$ ...
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Ideals of Polynomial Rings and Field Extensions

Let $F \subseteq K$ be fields, and suppose $f_1, ... , f_t \in F[X_1, ... , X_n]$. Let $R = F[X_1, ... , X_n]$, and let $S = K[X_1, ... , X_n]$. Is it always true that $(f_1S + \cdots + f_t S) \cap ...
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Knapp (Basic Algebra) Prop 8.52, error?

The above proposition says: Let $R$ be a Noetherian ring and let $I$ and $P$ be ideals of $R$ where $P$ is a prime ideal. If $IP=I$, then $I=0$. I feel that this is false. After passing to ...
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Checking whether an ideal is inside another.

I want to determine whether this ideals are contained in each other or not. 1) In $\mathbb{R}[x]$, $I_1= \langle x(1+x^2) \rangle$ and $I_2= \langle x^2(1+x) \rangle$. I can see that $I_1 \not\...
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A problem on maximal ideal in polynomial ring.

Let $\Bbb R[x]$ be the polynomial ring over $\Bbb R$ in one variable. Let $I\subseteq\Bbb R[x]$ be an ideal. Then which are true? $I$ is a maximal ideal if and only if $I$ is a non-zero prime ...
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60 views

Number of ideals in a finite-dimensional $K$-algebra

Let $K$ be a field and $A$ be a finite-dimensional $K$-algebra. Does $A$ have finitely many ideals? (I know that $A$ has finitely many prime ideals.)
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Intersection of a prime ideal with a non-prime ideal

Let $I_1$ be a prime ideal of $\mathbb{C}[X_1,...,X_n]$ and $I_2$ be another ideal of $\mathbb{C}[X_1,...,X_n]$ which is not a prime ideal. Assume further than $I_2 \not\subset I_1$. Under what ...
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110 views

Finding Ideals in $\begin{bmatrix} \mathbb{Q} & \mathbb{Q}\\ 0 & 0 \end{bmatrix}$

I am looking to find the left, right and two sided ideals of the ring R = $\begin{bmatrix} \mathbb{Q} & \mathbb{Q}\\ 0 & 0 \end{bmatrix}$. It is in finding the left ideals that I am stuck. ...
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The form of an element in $R[x]/I$ where $I$ is a principal ideal

It was stated in this question (Give an intuitive explanation for polynomial quotient ring, or polynomial ring mod kernel) that given a field $F$, a general element of $F[x]/(a_n x^n + \ldots + a_0)$ ...
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How can I visualize ideals on a ring of integers of imaginary quadratic fields?

If I were to visualize the ideal $(2, 3+3i)$ of $Z[i]$ on the complex plane, I would find a gcd of $2$ and $3+3i$ (for example $1+i$) and the ideal $(2, 3+3i)$ is identical to $(1+i)Z[i]$, which forms ...
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Checking that a set is a finitely generated ideal

The exercise asks us to prove that $I = \{ f \in \Bbb R[X,Y,Z] \mid f(a,b,c) = 0, ~\forall\,(a,b,c)\in \Bbb S^2 \}$ is a finitely generated ideal of $\Bbb R[X,Y,Z]$. Well, clearly $I$ is an ideal of $...