Euclidean ring

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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).
References in periodicals archive ?
Obviously, Z is a Euclidean Z-module since it is a Euclidean domain.
It is possible to make use of conventional spatial index structures for some high-dimensional Euclidean domains.
1), and we apply this result to plane domains with their Poincare metrics and Euclidean domains with their quasihyperbolic metrics (see respectively Theorems 3.