GAP (ResClasses) - Chapter 3: Semilocalizations of the Integers

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3 Semilocalizations of the Integers
3.1 Entering semilocalizations of the integers
3.1-1 Z_pi
3.2 Methods for semilocalizations of the integers

### 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

`‣ Z_pi`( pi ) | ( function )

`‣ Z_pi`( p ) | ( function )

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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