Articles of ring theory

When do we have $Rad(I)=I$ for an ideal $I$ of a ring $R$?

This is kind of a follow-up question about calculating the radical of an ideal. Since $Rad(I)$ is the intersection of all the prime ideals of $R$ that contain $I$, which is a property I learned from this article in wikipedia, we have that $$ Rad(I)=I $$ whenever $I$ is a prime ideal. My question is: […]

isomorphism, integers of mod $n$.

Hello I think this is true, but I’m not sure. Setup: If $n = p_{1}\cdot p_{2} \cdots p_{n}$ where $p_{i}$ prime for all $i\in\lbrace 1,\dots,n\rbrace$. Define the ring $A = p_{j}\mathbb{Z}/n\mathbb{Z}$. Question: Is $A$ isomorphic to $\mathbb{Z}/(n/p_{j})\mathbb{Z}$? And why?

Integral domain, UFD and PID related problem

(i) Let $R$ be an integral domain that has irreducible elements. Prove that $R[X]$ is not A PID. (ii) Let $R$ be a UFD and $K$ its field of fractions. Let $f \in R[X]$ be a monic polynomial with $\alpha$ a root of $f$ in $K$. Show that $\alpha \in R$. I think I could […]

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 $a_i$ […]

Extended ideals in power series ring

Let $A$ be a commutative ring with $1$ and consider the ring of formal power series $A[[X]]$. If $I \subseteq A$ is an ideal, let $I[[X]]$ denote the set of power series with coefficients in $I$. This is an ideal; it is the kernel of the reduction homomorphism $A[[X]] \to (A/I)[[X]]$. Let $IA[[X]]$ denote the […]

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 […]

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 […]

Trouble in understanding a proof of a theorem related to UFD.

This question already has an answer here: How can I show that $ab \sim \gcd (a,b) {\operatorname{lcm} (a,b)}$ for any $a,b \in R \setminus \{0\}$? 1 answer