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3 Semilocalizations of the Integers

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

3.1 Entering semilocalizations of the integers

3.1-1 Z_pi

Returns: the ring \(ℤ_{(\pi)}\) or the ring \(ℤ_{(p)}\), respectively.

The returned ring has the property IsZ_pi. The set pi of non-invertible 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 non-invertible 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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