# Tagged Questions

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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### 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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### 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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### 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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### 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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### $\mathfrak{a\subseteq b}$ 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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