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

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 $K_1[x_1,x_2,\dots,x_n]$?

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 […]

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

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 […]

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?

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?

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 […]

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 […]

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 […]

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