# 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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### Quotient cancellation for invertible ideals of orders in quadratic fields

Let $K/\mathbb{Q}$ be an imaginary quadratic field, $m\ge 1$ be a positive integer and let $\mathcal{O}=\mathbb{Z}+m\mathcal{O}_K\subset \mathcal{O}_K$ be the unique order of index (equivalently, ...
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### When the element-wise product of two ideals produces an ideal

Consider the ring $R=\mathbb C[X,Y]$. For every two ideals $I,J$ of $R$, define $I*J:=\{ij : i\in I, j\in J\}$. Now definitely, $I*J=J*I$ always holds. If $I$ is principal, then actually $I*J$ is an ...
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### ideal extension in polynomial ring over quotient field

This is probably easy, but I can't seem to figure it out. I have a $K$-algebra $B$, which is a domain and an irreducible polynomial $f \in K[x]$. Why does $B[x]/f$ embed in $Q(B)[x]/f$ where $Q(B)$ is ...
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### Representing an finite-dimensional algebra over a field using a quotient ring

My professor was talking about different ways to think about an algebra $A$ over a field $k$. One that she mentioned briefly but didn't go into much detail on was that of a quotient ring. Roughly the ...
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### Representing an algebra over a field using a quotient ring

My professor was talking about different ways to think about an algebra $A$ over a field $k$. One that she mentioned briefly but didn't go into much detail on was that of a quotient ring. Roughly the ...
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### Maximality of an ideal for showing that an algebra is in fact a field

I have an algebra $A$ over the field $F$, with the finite dimensionality $n$ as a vector space over $F$. I can also assume that $A$ is an integral domain. Assuming that $v_1,...,v_n$ is a spanning ...
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### If a ring is noetherian, then so is its subring

Suppose $A\subset R$ are rings so that $R$ is also an $A$-module. Assume further that there is an $A$-module homomorphism $R\to A$ that is identity on $A$. How to deduce that $A$ is noetherian if $R$ ...
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### Invertible elements of $\mathbb{Z}_3[x] / (x^4+x^3-1)^3$

Let $F=\mathbb{Z}/3\mathbb{Z}$, $h(x)=x^4+x^3-1$, $R = F[x]/(h(x)^3)$. I know $R$ has $4$ ideals and $1$ maximal ideal. Let $M$ be the maximal ideal $(h(x))/(h(x)^3)$ I need to find the number of ...
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### Is there a ring containing ideals where the union of the ideals forms a subring but not an ideal?

In general the union of two subrings is not a subring, and likewise for the union of two ideals. However, I was wondering if there exists an example of a union of ideals that is actually a subring but ...
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### If two ideals agree on $K\lhd R$ and yield the same quotients $\pi_K(I)=\pi_K(J)$, are they equal?

Inspired by my lectures on the basics of commutative algebra, I tried to investigate whether the noetherian property still holds when taking extensions (here meaning: if $I\lhd R$ and $R/I$ both ...
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### Does $\frac{b}{s} \in S^{-1} I$ directly implies that $b \in I$?

Let $R$ be a commutative ring with identity $1_R$, $0 \not \in S\subseteq R$ be a multiplicative set, and $I\subseteq R$ be an ideal of $R$.Consider the ring of quotients $S^{-1}I$. I was trying to ...
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### C* algebra exact sequences and ideals

if you have C* algebras $A,B$ and $C$ and $\exists$ a short exact sequence as follows $0\rightarrow A\rightarrow B \rightarrow C \rightarrow 0$ where the functions are $\phi$ and $\psi$ respectively, ...
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### Is the ideal generated by ${4,x}$ a principal ideal in $Z[x]$? [duplicate]

I've : $I=<p,x>$ is not a principal ideal in $Z[x]$ where p is prime. My question : Is $I=<p,x>$ a principal ideal in $Z[x]$ where p is not a prime? More particularly, is the ideal ...
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### Class Number Calculation of a Real Quadratic Number Field

I am looking at the example below. Can anyone explain how they end up with the contradiction. Why do they reduce $a^2-65b^2$ modulo $5$ to show that it has no integer solutions?
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### Showing that an ideal in $\mathbb{Z}[\sqrt{-21}]$ is principal

I have $\mathfrak{a}=(5,\sqrt{-21}-2).$ Can anyone tell me why $\mathfrak{a}^2$ is principal? I have multiplied the ideals out to obtain $$(5, 5\sqrt{-21}-10,-17-4\sqrt{-21})$$ How does this reduce ...
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### Showing that an ideal is not principal in $\mathbb{Z}[\sqrt{-21}]$

I am trying to show that the ideal $$(2,\sqrt{-21}-1)(3, \sqrt{-21})$$ is not principal in $\mathbb{Z}[\sqrt{-21}]$. Can anyone help with this?
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### If $I$ is a maximal ideal and $a\in R -I$, then the assumption that $I + (a) = R$ gives a contradiction

While I'm trying to prove that Let $S$ be a multiplicative set in the commutative ring $R$ with identity s.t $0 \not \in S$.Let $I$ be a maximal ideal in $S^c = R - S$. Then show that $I$ is a ...
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### Prime ideal containing an ideal and not intersecting a multiplicatively closed set [closed]

Suppose $R$ is a ring and $I$ an ideal of $R$, moreover suppose $M$ is a multiplicatively closed subset of $R$ such that $I$ does not intersect $M$ can you always find a prime ideal $I\subset P$ also ...
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### Fundamental Theorem on Homomorphisms - Application

I want to show 1.) For $n,m\in \mathbb Z$ with $m|n$ there exists a ring homomorphism between $\mathbb{Z}/n\mathbb{Z} \rightarrow \mathbb{Z}/m\mathbb{Z}$. I know that there is the canonical ...
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