[1X3. Semilocalizations of the Integers[0X This package implements residue class unions of the semilocalizations Z_(pi) of the ring of integers. It also provides the underlying [5XGAP[0X implementation of these rings themselves. [1X3.1 Entering semilocalizations of the integers[0X [1X3.1-1 Z_pi[0X [2X> Z_pi( [0X[3Xpi[0X[2X ) _______________________________________________________[0Xfunction [2X> Z_pi( [0X[3Xp[0X[2X ) ________________________________________________________[0Xfunction [6XReturns:[0X The ring Z_(pi) or the ring Z_(p), respectively. The returned ring has the property [10XIsZ_pi[0X. The set [3Xpi[0X of noninvertible primes can be retrieved by the operation [10XNoninvertiblePrimes[0X. [4X--------------------------- Example ----------------------------[0X [4X[0X [4Xgap> R := Z_pi(2);[0X [4XZ_( 2 )[0X [4Xgap> S := Z_pi([2,5,7]);[0X [4XZ_( 2, 5, 7 )[0X [4X[0X [4X------------------------------------------------------------------[0X [1X3.2 Methods for semilocalizations of the integers[0X There are methods for the operations [10Xin[0X, [10XIntersection[0X, [10XIsSubset[0X, [10XStandardAssociate[0X, [10XGcd[0X, [10XLcm[0X, [10XFactors[0X and [10XIsUnit[0X available for semilocalizations of the integers. For the documentation of these operations, see the [5XGAP[0X 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. [4X--------------------------- Example ----------------------------[0X [4X[0X [4Xgap> 4/7 in R; 3/2 in R;[0X [4Xtrue[0X [4Xfalse[0X [4Xgap> Intersection(R,Z_pi([3,11])); IsSubset(R,S);[0X [4XZ_( 2, 3, 11 )[0X [4Xtrue[0X [4X[0X [4X------------------------------------------------------------------[0X [4X--------------------------- Example ----------------------------[0X [4X[0X [4Xgap> StandardAssociate(R,-6/7);[0X [4X2[0X [4Xgap> Gcd(S,90/3,60/17,120/33);[0X [4X10[0X [4Xgap> Lcm(S,90/3,60/17,120/33);[0X [4X40[0X [4Xgap> Factors(R,840);[0X [4X[ 105, 2, 2, 2 ][0X [4Xgap> Factors(R,-2/3);[0X [4X[ -1/3, 2 ][0X [4Xgap> IsUnit(S,3/11);[0X [4Xtrue[0X [4X[0X [4X------------------------------------------------------------------[0X