Articles of ring theory

Why is $\mathbb{Z}$ an integral domain?

This question already has an answer here: Why is $\mathbb{Z}[\sqrt{-n}], n\ge 3$ not a UFD? 2 answers

If a polynomial divides another in a polynomial ring, will the division also occur in a subring?

Let $K_1\subset K_2$ be a field extension and let $p(x_1,x_2,\dots,x_n), q(x_1,x_2,\dots,x_n)\in K_1[x_1,x_2,\dots,x_n]$. If $p(x_1,x_2,\dots,x_n)\mid q(x_1,x_2,\dots,x_n)$ in $K_2[x_1,x_2,\dots,x_n]$, is it also true that $p(x_1,x_2,\dots,x_n)\mid q(x_1,x_2,\dots,x_n)$ in $K_1[x_1,x_2,\dots,x_n]$?

Characteristic of a product ring?

Let A and B be commutative rings with unity where char(A)=n and Char(B)=m s.t. n,m ∈ ℤ (and n,m ≠ 0). Prove or give counter example: if k ∈ ℤ+ and n,m both divide k, then Char (A x B)| k. Here was my thinking but I don’t know if it is true: Char(A x […]

A local Artinian ring with finite residue field is finite.

This question already has an answer here: Atiyah-Macdonald, Exercise 8.3: Artinian iff finite k-algebra. 2 answers

Prove that $\text{nil}(\overline{R})=\sqrt{A}/A$

Let $R$ be a commutative ring and $A \lt R$ an ideal. Define the radical of an ideal $A$ to be $\sqrt{A}:=\lbrace x \in R \mid x^n \in A \text{ for some } x \in \mathbb{Z}^+ \rbrace$. Let ${}^{-}:R \rightarrow R/A$ be the canonical ring homomorphism. Show that $\text{nil}(\overline{R})=\sqrt{A}/A$. I am a little confused by […]

Division in ring $Z$

Let $w=\frac{-1+\sqrt{-3}}2$,Find $q,r \in Z[w]$ such that $3+5w=(2-w)q+r$ What’s the best way to approach this kind of questions?

Comparison between rings and groups Question

Show by example, that for nonzero (fixed) elements a & b in a ring, the equation ax=b can have more than one solution. How does this compare to groups? Can someone help me compare rings to groups?

Left ideals of matrix rings are direct sum of column spaces?

Let $\mathbb K$ be a field and $M_n(\mathbb K)$ be the ring of the $n\times n$ matrices with entries in $\mathbb K$. Let $C_j\subset M_n(\mathbb K)$ be the subspace of all matrices which have all the entries zero except possibly those in the $j$-th column. How can I show that if $I$ is a left […]

Uniqueness of “Euclidean Expressions” in Euclidean Domains

In what follows let all variables be in some Euclidean Domain $R$. Suppose $n = kq + r$ s.t. $\deg(r) < \deg(k)$ or $r = 0$. Now suppose that $k \mid n$ so that $n = kq_1$ for some $q_1 \in R$. Question 1: Since we have that $n = kq + r = kq_1 […]

Which are integral domain

I know that $\mathbb Z[i]/n\mathbb Z[i]$ is an integral domain $\iff \langle n\rangle =n\mathbb Z[i]$ is a prime ideal of $\mathbb Z[i]\iff n$ is an prime element of $\mathbb Z[i].$ $2=(1+i)(1-i)$ where none of $1+i,1-i$ are units of $\mathbb Z[i].$ So $2$ is not irreducible and hence not prime. $13=(3+2i)(3-2i)$ where none of $3+2i,3-2i$ are […]