# Euclidean ring

## Euclidean ring

[yü‚klid·ē·ən ′riŋ]
(mathematics)
A commutative ring, together with a function, ƒ, from the nonzero elements of the ring to the nonnegative integers, such that (1) ƒ(xy) ≥ ƒ(x) if xy ≠ 0, and (2) for any members of the ring, x and y, with x ≠ 0, there are members q and r such that y = qx + r and either r =0 or f (r)<>f (x).
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Euclidean division is valid on Z[i]; hence, Z[i] is an Euclidean ring.
The skew polynomial ring [mathematical expression not reproducible] is left and right Euclidean ring whose left and right ideals are principal.
We know that [mathematical expression not reproducible] is left and right Euclidean ring whose left and right ideals are principal.
Let R[X] be the Euclidean ring of polynomials over the field of real numbers with the Euclidean function [phi](f) = deg(f) for each nonzero element f [member of] R[X].
Recall that for any nonzero element r in a Euclidean ring R with Euclidean function [phi], [phi]([1.

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