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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Why $\gcd(b,qb+r)=\gcd(b,r),\,$ so $\,\gcd(b,a) = \gcd(b,a\bmod b)$
Given: $a = qb + r$. Then it holds that $\gcd(a,b)=\gcd(b,r)$. That doesn't sound logical to me. Why is this so?
Addendum by LePressentiment on 11/29/2013: (in the interest of http://meta.math....
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Prove that $\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$
For all $a, m, n \in \mathbb{Z}^+$,
$$\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$$
Juan Liner
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Quotient ring of Gaussian integers
A very basic ring theory question, which I am not able to solve. How does one show that
$\mathbb{Z}[i]/(3-i) \cong \mathbb{Z}/10\mathbb{Z}$.
Extending the result: $\mathbb{Z}[i]/(a-ib) \cong \mathbb{...
user9413
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Classification of prime ideals of $\mathbb{Z}[X]$
Let $\mathbb{Z}[X]$ be the ring of polynomials in one variable over $\Bbb Z$.
My question: Is every prime ideal of $\mathbb{Z}[X]$ one of following types?
If yes, how would you prove this?
$(0)$.
$(...
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Why are maximal ideals prime?
Could anyone explain to me why maximal ideals are prime?
I'm approaching it like this, let $R$ be a commutative ring with $1$ and let $A$ be a maximal ideal. Let $a,b\in R:ab\in A$.
I'm trying to ...
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Euclid's Lemma $\,(a,b)=1,\ a\mid bc\Rightarrow a\mid c\,$ in Bezout, gcd, ideal form
Let $A$ be a UFD, e.g. $A = \Bbb Z$ or $\,\Bbb Q[x,y]$. Assume $a,b \in A$ are relatively prime, $c \in A$ and $a | bc$. To prove that $a|c$, is the following approach correct (or do you have to use ...
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$I+J=1 \Rightarrow I^n+J^n = 1$ for ideals (or elements in GCD domain) [Freshman's Dream Binomial Theorem]
Let $R$ be a commutative ring and $I_1, \dots, I_n$ pairwise comaximal ideals in $R$, i.e., $I_i + I_j = R$ for $i \neq j$. Why are the ideals $I_1^{n_1}, ... , I_r^{n_r}$ (for any $n_1,...,n_r \in\...
eddy3
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$\langle 2,x \rangle$ is a non-principal ideal in $\mathbb Z [x];\, $ $D[x]$ PID $\iff D$ field, for a domain $D$
Hi
I don't know how to show that $\langle 2,x \rangle$ is not principal and the definition of a principal ideal is unclear to me. I need help on this, please.
The ring that I am talking about is $\...
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What are the left and right ideals of matrix ring? How about the two sided ideals?
What are the left and right ideals of matrix ring? How about the two sided ideals?
user8195
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Ideals $\neq0$ in $F[x]$ & Euclidean domains are generated by elements $\neq0$ of minimal degree (Euclidean size), so Euclidean $\Rightarrow$ PID
Consider the following
Theorem Let $F$ be a field, $I$ a nonzero ideal in $F[x]$, and $g(x)$ an element of $F[x]$. Then, $I= \langle g(x)\rangle$ if and only if $g(x)$ is a nonzero polynomial of ...
user304120
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Structure of ideals in the product of two rings
$R$ and $S$ are two rings. Let $J$ be an ideal in $R\times S$. Then there are $I_{1}$, ideal of $R$, and $I_{2}$, ideal of $S$ such that $J=I_{1}\times I_{2}$.
For me it's obvious why $\left\{ r\in R\...
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When does the distributive law apply to ideals in a commutative ring?
Let $R$ be a commutative ring with identity and $I,J,K$ be ideals of $R$. If $I\supseteq J$ or $I\supseteq K$, we have the following modular law
$$ I\cap (J+K)=I\cap J + I\cap K$$
I was wondering if ...
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What do prime ideals in $k[x,y]$ look like?
Suppose that $k$ is an algebraically closed field. Then what do the prime ideals in the polynomial ring $k[x,y]$ look like?
As far as I know, the maximal ideals of $k[x,y]$ are of the form $(x-a,y-b)$...
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In a PID every nonzero prime ideal is maximal
In a principal ideal domain, prove that every non trivial prime ideal is a maximal ideal
Attempt: Let $R$ be the principal ideal domain. A principal ideal domain $R$ is an integral domain in which ...
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For ideals in a PID: sum = gcd, intersection = lcm
Suppose we have a Ring R which is a E.D, then it must be PID.
Suppose $I = (a)$ and $J =(b)$ are two ideals. Is it true that $I+J =(a)+(b) = (\gcd(a,b))?$
If this is true then how can we prove it ...
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