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 […]

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

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 […]

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 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 […]

Intereting Posts

Prove that for any element $b$, $|b|$ divides $|a|$ (order of $b$ divides order of $a$).
Sequence of convex functions
Is $\mathbb{R}/(P)$ isomorphic to $\mathbb C$ for every irreducible polynomial $P$ of degree $2?$
How would you solve the diophantine $x^4+y^4=2z^2$
Show that the unit sphere is strictly convex
Integral of $1/(x^2-1)^2$
Theory $T$ that is consistent, such that $ T + \mathop{Con}(T)$ is inconsistent
Integrating $x^2e^{-x}$ using Feynman's trick?
Does π start with two identical decimal sequences?
Calculate the slope of a line passing through the intersection of two lines
Why can I get two different results when calculating $\int \ln(z) \,dz$?
Proof for formula $\int e^{g(x)} dx = f(x) e^{g(x)}+C$
Non-circular proof of $\lim_{\theta \to 0}\frac{\sin\theta}{\theta} = 1$
Closed set in $\mathbb{R}^n$ is closure of some countable subset
Is this a perfect set?