Articles of ring homomorphism

Prove that the rings $End(\mathbb{Z}^{n})$ and $M_{n}(\mathbb{Z})$ are isomorphic

I need to prove that rings $End(\mathbb{Z}^{n})$ and $M_{n}(\mathbb{Z})$ are isomorphic. To start with, I let $A = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots& & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{pmatrix} \in M_{n}(\mathbb{Z})$ (so that all the […]

To show that $\langle x-a , y-b\rangle$ is a maximal ideal of $F$ by showing that $F/\langle x-a , y-b\rangle$ is a field

This question already has an answer here: $(x_1-a_1, x_2-a_2)$ is a maximal ideal of $K[x_1,x_2]$ [duplicate] 1 answer

How to “Visualize” Ring Homomorphisms/Isomorphisms?

I understand that the formal definition of a ring homomorphism is some function $f$ that maps $R$ to $S$ ($f: R \rightarrow S$) s.t. $f(ab) = f(a)f(b)$ and $f(a + b) = f(a) + f(b)$. But how do I “visualize” this? I guess I’m asking for a “dumbed down” explanation. For instance, I came across […]

What is an Homomorphism/Isomorphism “Saying”?

Outside of the technical definitions, what exactly is a homormorphism or an isomorphism “saying”? For instance, let’s we have a group or ring homomorphism $f$, from $A$ to $B$. Does a homomorphism mean that $f$ can send some $a_i$ in $A$ to $b_j$ in $B$, but has no way to “get it back”? Similarly, if […]

Let $R$ be a commutative Noetherian ring (with unity), and let $I$ be an ideal of $R$ such that $R/I \cong R$. Then is $I=(0)$?

Let $R$ be a commutative Noetherian ring (with unity), and let $I$ be an ideal of $R$ such that $R/I \cong R$. Then is it true that $I=(0)$ ? I know that surjective ring endomorphism on Noetherian ring is injective also, and since there is a natural surjection of $R$ onto $R/I$ so we get […]