Articles of abstract algebra

There are finitely many ideals containing $(a)$ in a PID

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 $a_i$ […]

Kernel of the homomorphism $\mathbb C → \mathbb C$ defined by $x→t,y→ t^{2},z→ t^{3}$.

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 $\left<z-x^3, y-x^2\right>$, […]

“Direct sums of injective modules over Noetherian ring is injective” and its analogue

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} $

“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 […]

Quotient of nilpotent group is nilpotent

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 […]

Prime elements of ring $\mathbb{Z}$

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

The Jacobson Radical of a Matrix Algebra

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 $\mathbb{Q} \simeq \mathbb{Q}$ as fields.

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 […]

Showing the polynomials form a Gröbner basis

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 […]

Does $\text{End}_R(I)=R$ always hold when $R$ is an integrally closed domain?

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 […]