  
  [1m[4m[31m2. Semilocalizations of the Integers[0m
  
  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  [1mGAP[0m library, they had to be
  implemented in this package.
  
  
  [1m[4m[31m2.1 Entering semilocalizations of the integers[0m
  
  [1m[4m[31m2.1-1 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).
  
  The  function  also accepts a single prime [22m[34mp[0m instead of the one-element list
  [22m[32m[22m[34mpi[0m = [ [22m[34mp[0m ][0m as argument.
  
  [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[35mgap> T := Z_pi([3,11]);[0m
    [22m[35mZ_( 3, 11 )[0m
    [22m[35m[0m
  [22m[35m------------------------------------------------------------------[0m
  
  [1m[4m[31m2.1-2 IsZ_pi[0m
  
  [1m[34m> IsZ_pi( [0m[22m[34mR[0m[1m[34m ) ______________________________________________________[0mproperty
  [1mReturns:[0m  [22m[32mtrue[0m  if  [22m[34mR[0m  is  a ring Z_(pi) for some set of primes pi and [22m[32mfalse[0m
            otherwise.
  
  [1m[4m[31m2.1-3 NoninvertiblePrimes[0m
  
  [1m[34m> NoninvertiblePrimes( [0m[22m[34mR[0m[1m[34m ) ________________________________________[0mattribute
  [1mReturns:[0m  The  set  of  noninvertible primes [22m[34mpi[0m in the semilocalization [22m[34mR[0m of
            the integers.
  
  
  [1m[4m[31m2.2 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 non-invertible
  prime factors of its numerator.
  
  [22m[35m---------------------------  Example  ----------------------------[0m
    [22m[35m[0m
    [22m[35mgap> 4/7 in R;[0m
    [22m[35mtrue[0m
    [22m[35mgap> 3/2 in R;[0m
    [22m[35mfalse[0m
    [22m[35mgap> U := Intersection(R,S,T);[0m
    [22m[35mZ_( 2, 3, 5, 7, 11 )[0m
    [22m[35mgap> IsSubset(R,U);[0m
    [22m[35mtrue[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
  
