GAP Manual: 5.3. DefaultRing

DefaultRing( r, s... )
DefaultRing( list )

In the first form DefaultRing returns the default ring that contains all the elements r, s... etc. In the second form DefaultRing returns the default ring that contains all the elements in the list list. If any element is not an element of a ring or if the elements lie in no common ring an error is raised.

The ring returned by DefaultRing need not be the smallest ring in which the elements lie. For example for elements from cyclotomic fields DefaultRing may return the ring of integers of the smallest cyclotomic field in which the elements lie, which need not be the smallest ring overall, because the elements may in fact lie in a smaller number field which is not a cyclotomic field.

For the exact definition of the default ring of a certain type of elements read the chapter describing this type.

DefaultRing is used by the ring functions like Quotient, IsPrime, Factors, or Gcd if no explicit ring is given.

    gap> DefaultRing( 1, -1 );
    Integers
    gap> DefaultRing( [10..20] );
    Integers 

Ring (see Ring) differs from DefaultRing in that it returns the smallest ring in which the elements lie, while DefaultRing may return a larger ring if that makes sense.