  
  [1m[4m[31m3. Semilocalizations of the Integers[0m
  
  In  this  package,  the semilocalizations Z_(pi) of the ring of integers are
  used as base rings for unions of residue classes.
  
  
  [1m[4m[31m3.1 Entering semilocalizations of the integers[0m
  
  [1m[4m[31m3.1-1 Z_pi[0m
  
  [1m[34m> Z_pi( [0m[22m[34mpi[0m[1m[34m ) _______________________________________________________[0mfunction
  [1m[34m> Z_pi( [0m[22m[34mp[0m[1m[34m ) ________________________________________________________[0mfunction
  [1mReturns:[0m  The ring Z_(pi) resp. the ring Z_(p).
  
  The  returned  ring  has  the  property  [22m[32mIsZ_pi[0m. The set [22m[34mpi[0m of noninvertible
  primes can be retrieved by the operation [22m[32mNoninvertiblePrimes[0m.
  
  [22m[35m---------------------------  Example  ----------------------------[0m
    [22m[35m[0m
    [22m[35mgap> R := Z_pi(2);[0m
    [22m[35mZ_( 2 )[0m
    [22m[35mgap> S := Z_pi([2,5,7]);[0m
    [22m[35mZ_( 2, 5, 7 )[0m
    [22m[35m[0m
  [22m[35m------------------------------------------------------------------[0m
  
  
  [1m[4m[31m3.2 Methods for semilocalizations of the integers[0m
  
  There   are   methods   for   the  operations  [22m[32min[0m,  [22m[32mIntersection[0m,  [22m[32mIsSubset[0m,
  [22m[32mStandardAssociate[0m,    [22m[32mGcd[0m,   [22m[32mLcm[0m,   [22m[32mFactors[0m   and   [22m[32mIsUnit[0m   available   for
  semilocalizations   of   the   integers.  For  the  documentation  of  these
  operations,  see  the  [1mGAP[0m  reference  manual.  The standard associate of an
  element  of  a  ring  Z_(pi)  is defined by the product of the noninvertible
  prime factors of its numerator.
  
  [22m[35m---------------------------  Example  ----------------------------[0m
    [22m[35m[0m
    [22m[35mgap> 4/7 in R; 3/2 in R;[0m
    [22m[35mtrue[0m
    [22m[35mfalse[0m
    [22m[35mgap> Intersection(R,Z_pi([3,11])); IsSubset(R,S);[0m
    [22m[35mZ_( 2, 3, 11 )[0m
    [22m[35mtrue[0m
    [22m[35m[0m
  [22m[35m------------------------------------------------------------------[0m
  
  [22m[35m---------------------------  Example  ----------------------------[0m
    [22m[35m[0m
    [22m[35mgap> StandardAssociate(R,-6/7);[0m
    [22m[35m2[0m
    [22m[35mgap> Gcd(S,90/3,60/17,120/33);[0m
    [22m[35m10[0m
    [22m[35mgap> Lcm(S,90/3,60/17,120/33);[0m
    [22m[35m40[0m
    [22m[35mgap> Factors(R,840);[0m
    [22m[35m[ 105, 2, 2, 2 ][0m
    [22m[35mgap> Factors(R,-2/3);[0m
    [22m[35m[ -1/3, 2 ][0m
    [22m[35mgap> IsUnit(S,3/11);[0m
    [22m[35mtrue[0m
    [22m[35m[0m
  [22m[35m------------------------------------------------------------------[0m
  
