Element of minimum norm among non-units in Euclidean ring is a universal side divisor

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Statement

Suppose R is a commutative unital ring and N is a Euclidean norm on R -- in particular, R is a Euclidean ring. Suppose b is a nonzero element of R that is not a unit, and such that N(b) \le N(r) for all nonzero non-units r of R. Then, b is a universal side divisor in R.

Definitions used

Euclidean norm

Further information: Euclidean norm

Universal side divisor

Further information: Universal side divisor

A nonzero non-unit b in a commutative unital ring R is termed a universal side divisor in R if for any a \in R, either b | a or there exists a unit r \in R such that b | a - r.

Related facts

Proof

Given: A commutative unital ring R with a Euclidean norm N. A nonzero non-unit element b of R such that N(b) \le N(r) for all nonzero non-units r in R.

To prove: b is a universal side divisor in R.

Proof: Pick any a \in R. Then, by the Euclidean algorithm, we can write:

a = bq + r

where either r = 0 or N(r) < N(b). If r = 0, b | a, and we are done. Otherwise, N(r) < N(b). By assumption, b has smallest norm among the non-units, so r must be a unit, hence bq = a - r, so b | a - r for a unit r, and we are done.

References

Textbook references