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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Jacobson radical and invertible element
Let $I_1,I_2$ be ideals of a ring $R$ such that $I_1+I_2=R$ and their intersection is contained in $J(R)$ (the Jacobson radical of $R$). Show that if $x_2$ is an element of $I_2$ s.t. $x_2+I_1$ is ...
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Equality of two completions
I have the following question.
Suppose $R$ is Noetherian ring, $I$ is ideal in $R$ and $S$ is multiplicatively closed set. Let $(I^n\colon\langle S\rangle) = \varphi^{-1}(I^nS^{-1}R),$ where $\varphi\...
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Differences in meaning for notations $\alpha_i$ versus $\alpha(i)$ and meaning of $\beta(ij)$ for denoting axioms in monomials
The following are partly taken from Malik and Sen's Fundamentals of Abstract Algebra
Background
First we note that we can reconstruct the monomial $x^\alpha=x_1^{\alpha_1}\cdots x_n^{\alpha_n}$ from $...
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Why are prime ideals proper?
As children we all learn this erroneous definition of a prime number: “a number $n\in \Bbb N$ is prime iff it’s only divided by one and itself”. Well that’s fine until the teacher asked us for ...
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Show that $(3,\sqrt [3]{11}+1)$ is a principal ideal in $\mathbb{Z}[\sqrt[3]{11}]$
I know how to verify $(3,\sqrt [3]{11}+1)=(\sqrt[3]{11}-2)$:
$(\sqrt[3]{11}-2)(\sqrt[3]{121}+2\sqrt[3]{11}+4)=11-8=3$ and then $\sqrt[3]{11}+1=\sqrt[3]{11}-2+3$
But if I don't know the answer, how can ...
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Question about an example on ring theory from Dummit and Foote
Background
Example: If $p$ is a prime, the ring $\Bbb{Z}[x]/p\Bbb{Z}[x]$ obtained by reducing $\Bbb{Z}[x]$ modulo the prime ideal $(p)$ is a Principal Ideal Domain, since the coeffiencets lie in the ...
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Consider in $Z[x]$ the polynomial $f(x)=x^2+x+1$. How do I determine whether $I=(2,f(x))$ is a prime ideal?
Consider in $Z[x]$ the polynomial $f(x)=x^2+x+1$. How do I determine whether $I=(2,f(x))$ is a prime ideal? I read that if a polynomial is primitive then the generated ideal is prime (Why?). So $(f(x))...
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Is the homomorphic image of a ring an ideal of the co domain? [closed]
Q. If f be a homomorphism from a ring R into a ring R'. Then show that f(R) is an ideal of R'.
As per my knowledge, it is not possible. I want a very clear idea about this question and the solution.
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How to break symmetry of a polynomial ideal to simplify Groebner basis?
I have an ideal $I$ generated by a set of polynomials $\{ p_i \}$. There are some variable permutations to which the ideal is symmetric. By this I mean (apologies if there is a standard term for this) ...
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Ideal $ \langle x_1^2 - x_1 , \ldots , x_n^2 - x_n \rangle $ radical?
Consider the ideal generated by the Boolean constraints
$$
P = \langle x_1^2 - x_1 , \ldots , x_n^2 - x_n \rangle.
$$
Is $P$ a radical ideal?
A few attempts. The above statement is supposed to be true ...
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Is $I_p=(p,x^2+1)$ a prime ideal of $\mathbb Z[x]$? What is the maximal ideals of $\mathbb Z[x]$ containing $I_p$ where $p=2,3,5$?
Let $I_p$ be the ideal of $\mathbb Z[x]$ generated by $p$ and $x^2+1$.
Problem: Is $I_p=(p,x^2+1)$ a prime ideal of $\mathbb Z[x]$? What is the maximal ideals of $\mathbb Z[x]$ containing $I_p$ where $...
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Isomorphism $\mathbb Z[\omega]/(1-\omega)^2\cong (\mathbb Z/(p))[x]/(1-X)^2$, $\omega$ is the $p-$th root of unity.
Im reading the following proof of Fermat's Last Theorem from Keith Conrad
https://kconrad.math.uconn.edu/blurbs/gradnumthy/fltreg.pdf
On page 5 he mentions that $\mathbb Z[\omega]/(1-\omega)^2\cong (\...
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Nilpotent Lie-Algebra $g$: $g^{i+1} ⊆ g^i$ ideal in $g$?
Assume $g$ to be a nilpotent Lie-Algebra.
Nilpotency means that we can find an index $n$ such that:
$g^n = \{0\}$
for the series defined as:
$g^0 = g$
$g^{i+1} = \operatorname{span}\{[g,g^i]\}$
Why is ...
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Nonunital non commutative ring with 3 ideals...
It is well known that if a (unital commutative) ring A has only three ideals ({0}, J, A), then the quotient A/J is a field.
But, what can we conclude about A/J if A is not commutative nor unital but ...
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Question about Finding the order of the quotient ring $\mathbb Z[\sqrt{19}]/I$
I have a doubt concerning the problem mentioned at Finding the order of the quotient ring $\mathbb{Z}[\sqrt{-19}]/I$.
In this post it's shown that $$
\mathbb{Z}[\sqrt{-19}]/I\cong \mathbb{Z}[X]/(X^{2}+...
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Find the ideal class group of $\mathbb{Q}(\sqrt{-5})$ by using the factorization theorem
Let $K=\mathbb{Q}(\sqrt {-5})$. We have shown that $\mathcal{O}_K$ has the integral basis $1,\sqrt{-5}$ and $D=4d=-20$. By computing the Minkowski's constant:$$M_K=\sqrt{|D|}\Big(\frac{4}{\pi}\Big)^{...
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the order of $R=\Bbb{Z}[x]/(ax+b, x^2+5)$ is $5a^2+b^2$
Let $a,b \in \Bbb{Z}$. When $a\neq 0$, I want to prove the order of $R=\Bbb{Z}[x]/(ax+b, x^2+5)$ is $5a^2+b^2$.
$R\cong \Bbb{Z}[-b/a]/((-b/a)^2+5)$. If I could prove the last ring is isomorphic to $\...
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Give two maximal ideals of a $\mathbb{Q}[x]$ s.t. the two quotient rings are not isomorphic.
A quick note on notation, $\mathbb{Q}[x]$ is the polynomial ring, and $\mathbb{F}_2$ is the field of two elements.
I had an exam and one of the questions was:
We say an ideal $I$ of a ring $R$ is ...
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Vakil's The Rising Sea Exercise 3.7.H (Version 2022)
The original exercise is on the page 127:
In $\mathbb{A}_n = \text{Spec}\ k[x_1,\dots,x_n]$, the subset cut
out by $f(x_1,\dots,x_n)\in k[x_1,\dots,x_n]$ should certainly have irreducible components ...
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Ideal generated by $\langle x^2+y^2-1,y-x^2+1\rangle$
Let $K$ be a field. While doing an exercise I am trying to find the ideal $I:=\langle x^2+y^2-1,y-x^2+1\rangle$ in $K[x,y]$. I am guessing that the ideal is principal since otherwise the exercise ...
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Is it possible for a ring to fail to have any immediately-submaximal ideals?
Let rings be commutative and unital.
Let an immediately-submaximal ideal be a non-maximal ideal $I$ such that, for all maximal ideals $K$ such that $I \subset K$, for every ideal $J$ such that $I \...
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Characterizations of the Jacobson Radical
I am currently studying the concept of the Jacobson radical of a ring, and have gotten confused about whether or not certain conditions are equivalent characterizations of the radical. Suppose that $\...
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What are the ideals of $n\mathbb{Z}$? [duplicate]
I know how to find the ideals of $\mathbb{Z}$. However, now I am trying to find the ideals of $n\mathbb{Z}$ for $n\in \mathbb{N}^+$. Using the same ideas about $\mathbb{Z}$, I have there questions:
...
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not irreducible but prime in a non-domain [duplicate]
Consider $\mathbf Z/6\mathbf Z$ as a ring.
It is not an integral domain since it contains zero-divisors, such as the element $[3]$ for example. Note that $[3]$ is not irreducible ($[3]^2 = [3]$), yet $...
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Equivalent definition for minimal ideals for commutative rings
Background
The following post on minimal ideals is a continuation and a counterpart to the following post on maxmial ideals. The quoted materials are taken from the following sources;
Fundamentals of ...
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How Should I show that these $k$-algebras are not Isomorphic?
Question
Show that the $k$-algebras $k[x,y]/\langle xy \rangle$ and $k[x,y]/\langle xy-1 \rangle$ are not isomorphic.
Attempt
At first, I thought $xy=0$. This would mean both $x$ and $y$ are zero ...
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How should I go about this proof about homogeneous polynomials?
Question
Let $f_1,…,f_s$ be homogeneous polynomials of total degrees $d_1<d_2\leq …\leq d_s$ and let $I=\langle f_1,\ldots,f_s\rangle\subseteq k$. Show that if $g$ is another homogeneous polynomial ...
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Universal property definition of an ideal generated by a subset?
I'm puzzled by the definition of ideals generated by a subset of a ring in Aluffi, Algebra: Ch 0.
The previous chapter on groups is (for an algebra book of this level) quite categorical in spirit. In ...
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Question about showing $(x,y)$ is a maximal ideal of $\Bbb{Q}[x,y]/F[x,y]$ [duplicate]
Background
Theorem 1: Let $M$ be an ideal in a commutative ring $R$ with identity $1_R$. Then $M$ is a maximal ideal if and only if the quotient ring $R/M$ is a field.
Exercsie 1: Prove that $(x)$ ...
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Binomial theorem for ideals
I was proving the statement that if $I$ and $J$ solvable ideals of Lie algebra $L$, then $I + J$ is a solvable ideal of $L$.
The proof is we know $$(I+J)/J\cong I/I\cap J.$$ Since $I,J$ are solvable ...
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Primitive idempotent and bilateral ideals
I'm trying to show for my algebra class that in a semisimple ring with unity $R$ (not necessarily commutative), every primitive idempotent element must belong to a minimal two-sided ideal.
Here, by ...
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Being explicit about the kernel of the map $R[x,y]\to \frac{R}{I}[x,y]$ and coefficients of $\frac{R}{I}[x,y]$
The following is taken from the text University Algebra by: N.S Gopalkrishnan
Background
Exercise 17: Let $I$ be an ideal of a commutative ring $R$ and let $I[x,y]$ consist of those polynomials with ...
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Alternative solution to showing that $\langle x^2 +1, y\rangle$ is a maximal ideal and its possible generalization?
The following is taken from the text An Introduction to Grobner Bases by: Ralf Froberg, and the following Notes: $\langle x^2 +1, y\rangle$ is maximal, pg.3 Question (5a)
Background
Notation 1: $\...
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What do the words "descends" and "Induced" mean in the following quoted passage?
The following is taken from pg 4 section 6.1 of the following notes
Background
$\quad$ We start by observing that a ring homomorphism descends to quotient rings whenever the image of an ideal on the ...
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Are determinantal ideals Cohen-Macaulay?
Let $R=K[X_{ij}:i=1,\dots,m,j=1,\dots,n]$. The ideal in $R$ generated by all the $t$-minors of the $m\times n$ matrix
$$ X=\begin{pmatrix}
X_{11} & X_{12} & \dots & X_{1n}\\
X_{21} &...
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Set of zero-divisors in a ring is a union of prime ideals [duplicate]
This is a problem from Atiyah-MacDonald Introduction to Commutative Algebra and it goes as follows:
In a ring $A$, let $\Sigma$ be the set of all ideals in which every element is a zero-divisor. Show ...
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Does the closure of product of two ideals satisfy $\overline{I_1I_2}=\overline{I_1}\ \overline{I_2}$.
Let $A$ be a $C^{\ast}$ algebra and $I_1$ and $I_2$ be two ideals in $A$. Is it true that $\overline{I_1I_2}=\overline{I_1}\ \overline{I_2}$?
It is clear that $\overline{I_1I_2} \supseteq \overline{...
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Ideal $\langle 3,x-1,y-2\rangle$ in $\mathbb{Z}[x,y]$
I am studying a little bit of ideals and come up with the exercise to show that the ideal $\langle 3,x-1,y-2\rangle$ is not equal to $\langle 1\rangle$ in the polynomial ring $\mathbb{Z}[x,y]$. At ...
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Understanding Radicals as Intersections of Prime Ideals and as the Preimage of a Nilradical
I ran into a proposition that the radical $r(a)$ equals the intersection of the prime ideals containing $a$. Then it is said that the latter can be understood through the following result: Let $R$ be ...
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the ideal $〈6〉$ in the ring $(\mathbb Z,+,.)$ [duplicate]
I am trying to solve this past exam question:
In the ring $(\mathbb Z,+,.)$, the ideal $〈6〉$ is
(a) maximal
(b) prime
(c) strongly prime
(d) another answer.
Which option is correct?
The only theorem I ...
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What is the semidirect product we use in Levi Decomposition
So using the Levi Decomposition for any lie algebra $\mathfrak{g}$, there exists a semisimple subalgebra $\mathfrak{s}$ such that:
$Rad(\mathfrak{g})$$\ltimes$$\mathfrak{s}$=$\mathfrak{g}$
However in ...
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If $R$ is an integral domain, $I$ is an ideal of $R$, and $0\neq f: I \to R$ is an $R$-module homomorphism, can we conclude that $f$ is injective?
If $R = \mathbb{Z}$, $0 \neq I \unlhd \mathbb{Z}$, and $0 \neq f: I \to \mathbb{Z}$ is an arbitrary $\mathbb{Z}$-module homomorphism, then $f$ must be injective.
This leads to the question:
If $R$ is ...
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$R = \mathbb R[X,Y]/(XY - 1)$ and $I$ be the ideal of $R$ generated by the image of the element $X - Y$ in $R$. Describe $R/I$
Let $R = \mathbb R[X,Y]/(XY - 1)$ ($\mathbb R$ is the set of real numbers) and I be the ideal of R generated by the image of the element X - Y in R.
I want to find a way to describe R/I, i.e. find a ...
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Mistake in Proof "Every unique factorization domain is a principal ideal domain"
While doing my Algebra HW, I "proved" that every unique factorization domain (UFD) is a principal ideal domain (PID). I know that this is not true, however I fail to see where exactly is ...
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If $J$ is a two-sided ideal of $k$-algebra $A\otimes_k B$, then $I=J\cap B$ is a two-sided ideal of $B$.
Let $A$ and $B$ be finite dimensional $k$-algebra, where $k$ is a field.
If $J$ is an two sided ideal of $k$-algebra $A\otimes_k B$, consider $I=J\cap B$, I stuck with proving that I is an two sided ...
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Show that $M_n(P) $ is a prime ideal of $M_n(R) $.
Let $R$ be a ring and $P$ be a prime ideals of $R$. Then $M_n(P) $ is a prime ideal of $M_n(R) $.
One proof of this I know is by using the fact that any ideal of $M_n(R) $ is of the form $M_n(I) $, ...
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Factoring ideals into prime ideals
I am currently working on a problem from the book “Introductory Algebraic Number Theory” by Kenneth S. Williams and Saban Alaca, and I would like to verify my solution.
The problem is:
Factor $<6&...
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why is the universal side divisor called universal?
With respect to the usage of the term "universal" in category theory I struggle to see the connection? In terms of elements, if we were to let divisibility represent the morphism, then ...
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Ring whose finitely generated ideals are principal [duplicate]
The boolean ring $\prod_{n\in N} (Z/2Z)$ is an example of rings that verifie two properties
every finitely generated ideals are principal
there exits ideals that are not principal.
My question : is ...
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Unital Ring $R$ (Commutativity Not Assumed) is a Field if and only if Maximal Ideal is $0$.
I have only seen this statement proven under the assumption that $R$ is commutative. However, what if we dropped this assumption? (And before somebody comments this, I am aware that the ring will have ...
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