5.8 IsIntegralRing

IsIntegralRing( R )

IsIntegeralRing returns true if the ring R is integral and false otherwise.

A ring R is called integral if it is commutative and if for all elements r and s of R we have r s = 0_R implies that either r or s is 0_R.

    gap> IsIntegralRing( Integers );
    true 

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

The default function called this way is RingOps.IsIntegralRing, which tests whether the product of each pair of nonzero elements is unequal to zero, unless R is infinite. This function is seldom overlaid, because most rings already have the flag bound.

Previous Up Top Next
Index