Let $R$ be a PID. If $(a)$ is a nonzero ideal, then there are finitely many ideals containing $(a)$. I know that this question has already been asked/answered here, but I wanted to write a more explicit solution. Is the following correct? Since $R$ is a UFD, let $a = a_1a_2\cdots a_n$ where each $a_i$ […]

I think we have $z-x^3$, $y-x^2$, and $z^2-y^3$ as elements of the kernel of the homomorphism $\mathbb C[x,y,z] → \mathbb C[t]$ deﬁned by $x→t,y→ t^{2},z→ t^{3}$. But why the kernel is not generated by all the 3 elements, and only by $z-x^3$, $y-x^2$? I think maybe it is because of $z^2-y^3$ is in $\left<z-x^3, y-x^2\right>$, […]

I have a commutative algebra class and I heard the theorem from the professor: Let $R$ be a Noetherian ring and $\{E_i : i\in I\}$ be a collection of injective $R$-modules then $\bigoplus_{i\in I} E_i$ is also injective. My questions are: How to prove that? My professor does not talk about the proof of it […]

“Find all generators of $ (\mathbb{Z}_{27})^{\times} $” My attempt is below. Since $ (\mathbb{Z}_{n})^{\times} $ is a cyclic if and only if $ n = 1, 2, 4, p^n, 2p^n $, $ (\mathbb{Z}_{27})^{\times} $ is cyclic. And the order of $ (\mathbb{Z}_{27})^{\times} $ is $ 3^3 – 3^2 = 18 $ by the Euler’s phi […]

Edit: I managed to rephrase my proof in a way that does not resort to coset multiplication. I think the resulting proof is better. I’ve added it as an answer below, while preserving the original question to avoid wrecking the context. In his book Finite Group Theory, section 1.D, Isaacs mentions without proof the following […]

Find prime elements of the ring $\mathbb{Z}[\sqrt {-21}]$. Please help with some ideas.

I am trying to solve the following question. Let $A$ be the algebra over $\mathbb{C}$ consisting of matrices of the form \begin{pmatrix} * & * & 0 & 0 \\ * & * & 0 & 0 \\ * & * & * & 0 \\ * & * & * & * \\ \end{pmatrix} […]

What are the conditions for integers $D_1$ and $D_2$ so that $\mathbb{Q}[\sqrt {D_1}] \simeq \mathbb{Q}[\sqrt {D_2}]$ as fields. Here $\mathbb{Q}[\sqrt {D}] := \{a + b \sqrt D \mid a,b \in \mathbb{Q} \}$ Really not sure where to begin with this sort of problem. I was thinking that I should split into cases where the integer […]

Let $A$ be an $m \times n$ real matrix in row echelon form and $I \subset \mathbb{R}[x_1,\dots,x_n]$ is an ideal generated by polynomials $p_i = \sum_{j = 1}^na_{ij}x_j$ with $1 \leq i \leq m$. Then the generators form a Gröbner basis for $I$ w.r.t. some monomial order. So I guess one should try the standard […]

Let $R$ be a commutative ring with identity, and $I\neq 0$ an ideal of $R$, I’m thinking how to calculate $\text{End}_R(I)$. I have proved that when $R$ is a integral domain, $\text{End}_R(I)=\{r\in\text{Frac}(R)\mid rI\subset I\}$. So I’m curious about if $\text{End}_R(I)=R$ always holds when $R$ is also integrally closed. Can anyone help me with this? Thanks […]

Intereting Posts

Open set as a countable union of open bounded intervals
Is there a general formula for solving 4th degree equations (quartic)?
Find the radius of convergence of $\sum_{n=1}^{\infty}{n!x^{n!}}$
Integral $\int_0^{\pi/4}\log \tan x \frac{\cos 2x}{1+\alpha^2\sin^2 2x}dx=-\frac{\pi}{4\alpha}\text{arcsinh}\alpha$
A digital notebook for Mathematics?
How to show that the geodesics of a metric are the solutions to a second-order differential equation?
RSA when N=pq and p = q
Is there a simple proof of the fact that if free groups $F(S)$ and $F(S')$ are isomorphic, then $\operatorname{card}(S)=\operatorname{card}(S')?$
How to prove $2\sqrt{2+\sqrt{3}}=\sqrt{2}+\sqrt{6}$?
How to proof equivalent condition of algebra morphism and coalgebra morphism about Hopf algebra
Equivalence relations on classes instead of sets
Compute the weights of a $(\mathbb C^*)^{m+1}$-action on $H^0(\mathbb P^m, \mathcal O_{\mathbb P^m}(1))$
small o(1) notation
The map $\lambda: H^*(\tilde{G}_n)\to H^*(\tilde{G}_{n-1})$ maps Pontryagin classes to Pontryagin classes; why?
Why do books titled “Abstract Algebra” mostly deal with groups/rings/fields?