PID not implies Euclidean
This article gives the statement and possibly, proof, of a non-implication relation between two commutative unital ring properties. That is, it states that every commutative unital ring satisfying the first commutative unital ring property need not satisfy the second commutative unital ring property
View a complete list of commutative unital ring property non-implications | View a complete list of commutative unital ring property implications |Get help on looking up commutative unital ring property implications/non-implications
|
Statement
There exist principal ideal domains that are not Euclidean.
Facts used
- Euclidean ring that is not a field has a universal side divisor
- PID need not have a universal side divisor
Proof
The proof follows from facts (1) and (2). A specific example of a PID that does not have a universal side divisor is:
![{\displaystyle \mathbb {Z} \left[{\frac {1+{\sqrt {-19}}}{2}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb17dc804d1adeb142f5d624f5a59ef5d419c61a)
Further information: PID need not have a universal side divisor
References
Textbook references
Journal references
- A Principal Ideal Ring that is not a Euclidean ring by Jack C. Wilson, Math. Mag., pp.34-38