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Part CCSIR NET June 2024quotient-of-a-pid-is-principal-but-not-a-domain

Quotient of a pid is principal but not a domain

Let R be a principal ideal domain with a unique maximal ideal. Which of the following statements are necessarily true?

  1. A.Every quotient ring of R is a principal ideal domain
  2. B.There exists a quotient ring S of R and an ideal I ⊆ S which is not principal
  3. C.R has countably many ideals✓ Correct
  4. D.Every quotient ring S(≠ {0}) of R has a unique maximal ideal which is principal✓ Correct

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests property not inherited.

See pricing

50 are analysed free — try those first.

The trap it tests

Property not inherited

A property assumed to pass to subobjects, quotients, or through a chain. It does not.

Drill statements like this

Related counterexample: Every integral domain is a UFD

More on this topic

The chapter behind this: The ring hierarchy and where each inclusion is strict — free to read

From Rings and Fields › Euclidean, PID, UFD hierarchy

Last revised . Found a mistake? Tell us — corrections are the fastest thing we act on.

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