Semilocalizations of the Integers
gap> SizeScreen([75,24]);;

This package implements residue class unions of the semilocalizations \mathbb{Z}_{(\pi)} of the ring of integers. It also provides the underlying &GAP; implementation of these rings themselves.

Entering semilocalizations of the integers
The ring \mathbb{Z}_{(\pi)} or the ring \mathbb{Z}_{(p)}, respectively.
The returned ring has the property IsZ&uscore;pi. The set pi of noninvertible primes can be retrieved by the operation NoninvertiblePrimes. IsZ&uscore;pi NoninvertiblePrimes gap> R := Z_pi(2); Z_( 2 ) gap> S := Z_pi([2,5,7]); Z_( 2, 5, 7 )
Methods for semilocalizations of the integers
StandardAssociate
Gcd
Lcm
Factors
IsUnit

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 \mathbb{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