Let A be a Noetherian ring, then show that any ring empimorphism is also injective.

Hint: consider a chain of ideals

Solution:

As A is Noetherian, the chain of ideals becomes stationary, i.e for some n

Moreover we know that is also onto.

Suppose , then equals for some

Now as i.e

My question is why equals for some ?????? how do we get this?

and this line as i.e

Thanks