A commutative ring with identity is called a chain ring if all its ideals form a chain under inclusion. I want to know is there any proof for the fact that a zero dimensional local ring is a chain ring whenever its maximal ideal is principal?
A famous theorem by Kaplansky says that a commutative ring is a principal ideal ring iff all of its prime ideals are principal. By using a zero-dimensional local ring with a principal maximal ideal, you are in that situation.
A commutative, local principal ideal ring is well-known to be a chain ring (a.k.a. uniserial ring) as discussed in the wiki.