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
Could you please help me solve this question?
Let I1, I2 be ideals of a ring R (not necessarily commutative) s.t.
I1 + I2 = R
I1∩I2 = J(R) (where J(R) is the Jacobson radical)
Show that if x2 is an ...
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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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