A theorem of Hungerford (https://msp.org/pjm/1968/25-3/pjm-v25-n3-p11-p.pdf) states: Every PIR is a homomorphic image of a finite direct product of PID's. In the end, the author states a corollary, that a PIR is a finite direct product of PIDs iff the PIR has no non-zero nilpotent. My question is: Can we characterize those PIRs which are homomorphic image of a PID ?
2026-03-31 22:51:06.1774997466
Which principal ideal rings are homomorphic image of principal ideal domain?
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Yes, I think so.
If it is a proper quotient of a PID, then it is a $0$ dimensional Noetherian ring (actually quasi-Frobenius, even), which by virtue of its Artinianness is a special principal ring.
Conversely, a special principal ring (I am led to believe from the wiki) is a quotient of a discrete valuation ring, which is of course a PID.