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.
‣ Z_pi( pi ) | ( function ) |
‣ Z_pi( p ) | ( function ) |
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 )
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
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