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).