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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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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) ...
PPenguin's user avatar
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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 ...
Alexandros's user avatar
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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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1 answer
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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)^{...
Bowei Tang's user avatar
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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 ...
ettolrach's user avatar
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1 answer
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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 ...
Flynn Fehre's user avatar
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Section lying in subsheaf is a closed condition

I have the following problem. Suppose that $X$ is an integral projective variety over field $k$, $\mathcal{K}$ is a sheaf of fractions of structure sheaf $\mathcal{O}_X$, $\mathcal{F}$ is a locally ...
abcd1234's user avatar
1 vote
1 answer
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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 $\...
Jackson Wilson's user avatar
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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 ...
Mr Prof's user avatar
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1 answer
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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 ...
David M's user avatar
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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 $(x)$ ...
Seth's user avatar
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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 ...
Seth's user avatar
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1 answer
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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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Mistake in Faugère's improved F4 algorithm?

So I’m currently writing about the F4 algorithm and mainly work with the paper in which the algorithm was first published (any suggestions for other good sources are welcome). Now not only is the ...
user1315365's user avatar
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1 answer
48 views

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{...
Math Lover's user avatar
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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 ...
Aristarchus_'s user avatar
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1 answer
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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 ...
gbd's user avatar
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2 votes
2 answers
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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 ...
Albi's user avatar
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3 votes
1 answer
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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 ...
Liang Chen's user avatar
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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 ...
Jishnu's user avatar
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6 votes
1 answer
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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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2 votes
1 answer
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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) $, ...
Math Lover's user avatar
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1 vote
0 answers
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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&...
Aseel .A's user avatar
1 vote
1 answer
55 views

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 ...
DoubleA Batteries's user avatar
1 vote
0 answers
28 views

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 ...
wilhelm's user avatar
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2 votes
1 answer
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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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3 votes
2 answers
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On domain in which all right finitely generated ideals are principal

Let $R$ - a domain in which all right finitely generated ideals are principal. I want to prove that $aR \cap bR \ne 0$ for all $a, b \ne 0$. My idea is the following. Let $(a)$ and $(b)$ - two right ...
Irene's user avatar
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1 answer
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LAnn$_R(a)$ denotes the kernel of some ring homomorphism.

Let LAnn$_R(a)$ denote the collection of all left annihilators for arbitrary element $a\in R$. I'm curious to show that this subset LAnn$_R(a)\subset R$ denotes an ideal specifically by finding a ring ...
JAG131's user avatar
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0 answers
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Are there any subsets $I$ of $\mathbb{Z}$ containing $0$ & closed under multiplication by elements of $\mathbb{Z}$, but not an ideal of $\mathbb{Z}$?

Are there any subsets $I$ of $\mathbb{Z}$ containing $0$ & closed under multiplication by elements of $\mathbb{Z}$, but not an ideal of $\mathbb{Z}$? This was an exam question at my university ...
SWright's user avatar
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1 vote
1 answer
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A commutative ring with unity which is also reduced having exactly two minimal prime ideals.

Let $R$ be a commutative reduced ring with unity. I want to find an example of such a ring which contains exactly two minimal prime ideals. For example if we take $\mathbb{Z_6}$ then $\langle2\rangle$ ...
Chaudhary's user avatar
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0 votes
1 answer
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If $I$ is a two-sided ideal of $R$ and $M$ is a left module over $R$, why is $IM$ also a left module over $R$? There is also a related question.

Let $R$ be a ring with unity. If $I$ is a two-sided ideal of $R$ and $M$ is a left module over $R$, then $IM$ is also a left module over $R$, where $IM = \lbrace {\sum_{i=1}^n a_im_i|a_i\in I, m_i\in ...
Liang Chen's user avatar

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