5.10 IsEuclideanRing

IsEuclideanRing( R )

IsEuclideanRing returns true if the ring R is a Euclidean ring and false otherwise.

A ring R is called a Euclidean ring if it is an integral ring and there exists a function delta, called the Euclidean degree, from R-{0_R} to the nonnegative integers, such that for every pair r in R and s in R-{0_R} there exists an element q such that either r - q s = 0_R or delta(r - q s) < delta( s ). The existence of this division with remainder implies that the Euclidean algorithm can be applied to compute a greatest common divisor of two elements, which in turn implies that R is a unique factorization ring.

    gap> IsEuclideanRing( Integers );
    true 

IsEuclideanRing first tests whether the flag R.isEuclideanRing is bound. If the flag is bound, it returns this value. Otherwise it calls R.operations.IsEuclideanRing( R ), remembers the returned value in R.isEuclideanRing, and returns it.

The default function called this way is RingOps.IsEuclideanRing, which just signals an error, because there is no generic way to test whether a ring is a Euclidean ring. This function is seldom overlaid because most rings already have the flag bound.

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