Noetherian ring, any empimorphism is also injective,

Let A be a Noetherian ring, then show that any ring empimorphism $\displaystyle \varphi A\to A$ is also injective.

Hint: consider a chain of ideals $\displaystyle ker\varphi\subset ker\varphi^2....$

Solution:

As A is Noetherian, the chain of ideals becomes stationary, i.e $\displaystyle ker\varphi\subset ker\varphi^2\subset....\subset ker\varphi^n=ker\varphi^{n+1}$ for some n

Moreover we know that $\displaystyle \varphi^{n}:A\to A$ is also onto.

Suppose $\displaystyle a\in ker\varphi$, then $\displaystyle a\in A$ equals $\displaystyle \varphi^{n}(b)$ for some $\displaystyle b\in A$

Now $\displaystyle \varphi(a)=0$ as $\displaystyle a\in ker\varphi\Rightarrow \varphi^{n+1}(b)=0\Rightarrow \varphi^{n}(b)=0$ i.e $\displaystyle a=0\Rightarrow ker\varphi=0$

My question is why $\displaystyle a\in A$ equals $\displaystyle \varphi^{n}(b)$ for some $\displaystyle b\in A$?????? how do we get this?

and this line $\displaystyle \varphi(a)=0$ as $\displaystyle a\in ker\varphi\Rightarrow \varphi^{n+1}(b)=0\Rightarrow \varphi^{n}(b)=0$ i.e $\displaystyle a=0\Rightarrow ker\varphi=0$

Thanks

Re: Noetherian ring, any empimorphism is also injective,

we know that φ^{n} is onto.

for suppose a is any element of A. since φ is onto, there exists b_{1} with φ(b_{1}) = a.

since b_{1} is likewise an element of A, there exists b_{2} with φ(b_{2}) = b_{1}, therefore φ^{2}(b_{2}) = φ(φ(b_{2})) = φ(b_{1}) = a.

a short inductive proof then shows that φ^{n} is onto, for any n.

so if a is any element of the kernel of φ (so that φ(a) = 0), we know there exists b in A with φ^{n}(b) = a, no matter what n we pick.

in particular, if we pick n = N (the integer at which the kernels stabilize), then a = φ^{N}(b).

in which case φ^{N+1}(b) = φ(φ^{N}(b)) = φ(a) = 0.

so if a is in ker(φ), then b is in ker(φ^{N+1}) = ker(φ^{N}). which means φ^{N}(b) = 0, that is: a = 0, so φ is injective.

Re: Noetherian ring, any empimorphism is also injective,

thank you, your reformulation of the problem helped me to finally get this :)