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### 2. Semilocalizations of the Integers

In the following the semilocalizations Z_(pi) of the ring of integers are needed as base rings for unions of residue classes. Since these rings are not already implemented as domains in the GAP library, they had to be implemented in this package.

#### 2.1 Entering semilocalizations of the integers

##### 2.1-1 Z_pi

`> Z_pi`( pi ) | ( function )

`> Z_pi`( p ) | ( function )

Returns: The ring Z_(pi).

The function also accepts a single prime p instead of the one-element list `pi = [ p ]` as argument.

    gap> R := Z_pi(2);
    Z_( 2 )
    gap> S := Z_pi([2,5,7]);
    Z_( 2, 5, 7 )
    gap> T := Z_pi([3,11]);
    Z_( 3, 11 )

##### 2.1-2 IsZ_pi

`> IsZ_pi`( R ) | ( property )

Returns: `true` if R is a ring Z_(pi) for some set of primes pi and `false` otherwise.

##### 2.1-3 NoninvertiblePrimes

`> NoninvertiblePrimes`( R ) | ( attribute )

Returns: The set of noninvertible primes pi in the semilocalization R of the integers.

#### 2.2 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;
    true
    gap> 3/2 in R;
    false
    gap> U := Intersection(R,S,T);
    Z_( 2, 3, 5, 7, 11 )
    gap> IsSubset(R,U);
    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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