# Tagged Questions

This tag is for questions about rings, which are a type of algebraic structure studied in abstract algebra and algebraic number theory.

90 views

78 views

### Example of ring $R$ with ideals $I\neq J$ such that $R/I \cong R/J$ as modules

It's easy to prove that if $I$, $J$ are two-sided ideals and $R/I\cong R/J$ as modules over $R$, then $I=J$. What about left ideals? Is there a simple counterexample? I believe I've found an answer, ...
232 views

### A question about algebraically closed fields

A field $\mathbb{K}$ is said to be algebraically closed in practice if every polynomial over $\mathbb{K}$ of positive degree less than or equal to $10^{10}$ has zero belonging $\mathbb{K}$. The ...
347 views

### Finitely generated ideal in Boolean ring; how do we motivate the generator?

This problem is Exercise 11.3 in Atiyah/Macdonald Commutative Algebra. They ask to prove every finitely generated ideal in a Boolean ring is in fact a principal ideal. The question has been answered ...
39 views

### My question is about the definition of a map called the “reduction map”.

Let $G$ be a group and $N$ normal in $G$. I have read about a map $\alpha : G\rightarrow \frac{G}{N}$ called the reduction map mod $N$. I would love if someone could please explain this to me. Is it ...
142 views

### If for any two principal ideals one contains another, then for any two ideals one ideal contains another

Let $R$ be a commutative ring with identity. Assume that for any two principal ideals $Ra$ and $Rb$ we have either $Ra\subseteq Rb$ or $Rb\subseteq Ra$. Show that for any two ideals $I$ and $J$ in $R$,...
53 views

### ideal,ring,flat module,modules over R

Is there a characterization of modules (AND equivalent characterizations of rings R) over integral domains R with the property that each left ideal in R is flat?When all left ideals are projective,...
81 views

### A Question about the Proof of Eisenstein's Irreducibity Criterion

Statement: Let $f(x) = a_n x^n + a_{n-1} x^{n-1}+ \cdots + a_0 \in \mathbb Z[x]$. If there is a prime $p$ such that $p \nmid a_n, p \mid a_{n-1}, \dots,p \mid a_0$ and $p^2 \nmid a_0$, then $f(x)$ ...
67 views

### When does $n$-dimensional algebra have $m$-dimensional faithful representation?

Suppose we have an $n$-dimensional associative unital algebra $A$ over a field $k$ (assume $\operatorname{char}(k)=0$ and maybe even $k$ is closed). I would like to know what is the minimal ...
30 views

49 views

### A ring with prime characteristic

Let $p$ be a prime and let $R$ be a commutative ring with characteristic $p$. Prove that the number of elements of the set $$S_k=\{x\in R\;\lvert \;x^p=k\}\quad \text{for} \quad k\in \{1,2,\dots,p\}$$...
129 views

### Is every local ring the localization of some other ring?

One way of constructing a local ring is to start with any commutative ring, and localize all the elements outside of some maximal ideal (i.e., adjoining inverses to all those elements). But I'm ...
167 views

615 views

### Finitely generated ring.

Let $R$ be a finitely generated ring. Then is it true that $R^2$ is also finitely generated? My Attempt: I do not find a counterexample. I think it is true. Please someone help me to prove that $R^2$...