A surjective ring homomorphism can map a local ring to a local ring

Question

This question is from Chapter 2 of Lang's Algebra. Let f be a surjective ring homomorphism mapping ring A to ring A', assume A is local and A' is not zero ring. How to show A' is also local?

I am trying to solve this question by directly using the definition of local ring, i.e. the local ring is a commutative ring with only one maximal ideal. However, I don't know how to proceed further..

Thank you so much for your hints!

Answer 1

Hint: the first isomorphism theorem says $A/I\cong A’$ for some ideal $I$ of $A$. Then there is ideal correspondence...