### 3 Semilocalizations of the Integers

This package implements residue class unions of the semilocalizations ℤ_(π) 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_(π) 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_(π) 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