<!-- #################################################################### --> <!-- ## ## --> <!-- ## z_pi.xml ResClasses documentation Stefan Kohl ## --> <!-- ## ## --> <!-- ## $Id: z_pi.xml,v 1.21 2007/09/26 14:38:27 stefan Exp $ ## --> <!-- ## ## --> <!-- #################################################################### --> <Chapter Label="ch:Z_pi"> <Heading>Semilocalizations of the Integers</Heading> <Ignore Remark="set screen width to 75, for the example tester"> <Example> <![CDATA[ gap> SizeScreen([75,24]);; ]]> </Example> </Ignore> This package implements residue class unions of the semilocalizations <M>\mathbb{Z}_{(\pi)}</M> of the ring of integers. It also provides the underlying &GAP; implementation of these rings themselves. <!-- #################################################################### --> <Section Label="sec:DefiningZ_pi"> <Heading>Entering semilocalizations of the integers</Heading> <ManSection> <Func Name="Z_pi" Arg="pi" Label="by set of noninvertible primes"/> <Func Name="Z_pi" Arg="p" Label="by noninvertible prime"/> <Returns> The ring <M>\mathbb{Z}_{(\pi)}</M> or the ring <M>\mathbb{Z}_{(p)}</M>, respectively. </Returns> <Description> The returned ring has the property <C>IsZ&uscore;pi</C>. The set <A>pi</A> of noninvertible primes can be retrieved by the operation <C>NoninvertiblePrimes</C>. <Index Key="IsZ_pi"><C>IsZ&uscore;pi</C></Index> <Index Key="NoninvertiblePrimes" Subkey="of a semilocalization of Z"> <C>NoninvertiblePrimes</C> </Index> <Example> <![CDATA[ gap> R := Z_pi(2); Z_( 2 ) gap> S := Z_pi([2,5,7]); Z_( 2, 5, 7 ) ]]> </Example> </Description> </ManSection> </Section> <!-- #################################################################### --> <Section Label="sec:MethodsForZ_pi"> <Heading>Methods for semilocalizations of the integers</Heading> <Index Key="StandardAssociate" Subkey="of an element of a semilocalization of Z"> <C>StandardAssociate</C> </Index> <Index Key="Gcd" Subkey="of elements of a semilocalization of Z"> <C>Gcd</C> </Index> <Index Key="Lcm" Subkey="of elements of a semilocalization of Z"> <C>Lcm</C> </Index> <Index Key="Factors" Subkey="of an element of a semilocalization of Z"> <C>Factors</C> </Index> <Index Key="IsUnit" Subkey="for an element of a semilocalization of Z"> <C>IsUnit</C> </Index> There are methods for the operations <C>in</C>, <C>Intersection</C>, <C>IsSubset</C>, <C>StandardAssociate</C>, <C>Gcd</C>, <C>Lcm</C>, <C>Factors</C> and <C>IsUnit</C> 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 <M>\mathbb{Z}_{(\pi)}</M> is defined by the product of the noninvertible prime factors of its numerator. <Example> <![CDATA[ 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 ]]> </Example> <Example> <![CDATA[ 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 ]]> </Example> <Alt Only="HTML"> </Alt> </Section> <!-- #################################################################### --> </Chapter> <!-- #################################################################### -->