# 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).
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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Euclidean division is valid on Z[i]; hence, Z[i] is an Euclidean ring. All units in Z[i] are 1, -1, i, -i.
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.
They are neither left nor right Euclidean rings. However, left and right divisor can be defined for some suitable elements.
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.sub.R]) [less than or equal to] [phi](r), and [phi](r) = [phi]([1.sub.R]) if and only if r is a unit in R.

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