3. Semilocalizations of the Integers

In this package, the semilocalizations Z_(pi) of the ring of integers are used as base rings for unions of residue classes.

3.1 Entering semilocalizations of the integers

3.1-1 Z_pi

Returns: The ring Z_(pi) resp. the ring Z_(p).

The returned ring has the property IsZ_pi. The set pi of noninvertible primes can be retrieved by the operation NoninvertiblePrimes.

gap> R := Z_pi(2);
Z_( 2 )
gap> S := Z_pi([2,5,7]);
Z_( 2, 5, 7 )

3.2 Methods for semilocalizations of the integers

There are methods for the operations in, Intersection, IsSubset, StandardAssociate, Gcd, Lcm, Factors and IsUnit available for semilocalizations of the integers. For the documentation of these operations, see the GAP reference manual. The standard associate of an element of a ring Z_(pi) is defined by the product of the noninvertible prime factors of its numerator.

gap> 4/7 in R; 3/2 in R;
true
false
gap> Intersection(R,Z_pi([3,11])); IsSubset(R,S);
Z_( 2, 3, 11 )
true

gap> StandardAssociate(R,-6/7);
2
gap> Gcd(S,90/3,60/17,120/33);
10
gap> Lcm(S,90/3,60/17,120/33);
40
gap> Factors(R,840);
[ 105, 2, 2, 2 ]
gap> Factors(R,-2/3);
[ -1/3, 2 ]
gap> IsUnit(S,3/11);
true

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