# When is a zero dimensional local ring a chain ring?

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?

• This is proved in the paper : Hungerford, Thomas W. On the structure of principal ideal rings. Pacific J. Math. 25 1968 543–547. – Matthieu Romagny Jun 4 '18 at 15:22

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.

• Is its proof difficult? – Artor Waxsess Jun 5 '18 at 4:58
• @ArtorWaxsess Of the Cohen-Kaplansky theorem? I'm not familiar with the whole proof. I don't think it is too hard though. Check out Irving Kaplansky, Elementary divisors and modules, Trans. Amer. Math. Soc. 66 (1949), 464–491. MR 0031470 (11,155b). I'm not up to speed on the details for the second half, either. But they are both well-known and I suspect have been streamlined. And, for that matter, what I'm suggesting may be overkill, and there could be an alternative approach. This is just the first method I thought of. – rschwieb Jun 5 '18 at 13:18