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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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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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 ...
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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?
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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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