Localize Rings The package &LocalizeRingForHomalg; defines the classes of local(ized) rings, local ring elements and local matrices. These three objects can be used as data structures defined in &MatricesForHomalg; on which the &homalg; project can rely to do homological computations over localized rings.

A &homalg; local ring element contains two &homalg; ring elements, a numerator (&see; ) and a denominator (&see; ). A &homalg; local matrix contains a global &homalg; matrix as a numerator (&see; ) and a ring element as a denominator (&see; ). New constructors for ring elements and matrices are and in addition to the standard contructors introduced in other packages of the &homalg; project.

The local rings most prominently can be used with methods known from general &homalg; rings. The methods for doing the computations are presented in the appendix (), since they are not for external use. The new attributes and operations are documented here.

Since the objects inplemented here are representations from objects elsewhere in the &homalg; project (i.e. &MatricesForHomalg;), we want to stress that there are many other operations in &homalg;, which can be used in connection with the ones presented here. A few of them can be found in the examples and the appendix of this documentation.
Category and Representations <#Include Label="IsHomalgLocalRingRep"> <#Include Label="IsHomalgLocalRingElementRep"> <#Include Label="IsHomalgLocalMatrixRep">
Rings: Attributes <#Include Label="GeneratorsOfMaximalLeftIdeal"> <#Include Label="GeneratorsOfMaximalRightIdeal">
Operations and Functions <#Include Label="AssociatedGlobalRing:ring"> <#Include Label="AssociatedGlobalRing:element"> <#Include Label="AssociatedGlobalRing:matrix"> <#Include Label="Numerator:element"> <#Include Label="Numerator:matrix"> <#Include Label="Denominator:element"> <#Include Label="Denominator:matrix"> <#Include Label="Name"> <#Include Label="SetMatElm"> <#Include Label="AddToMatElm"> <#Include Label="MatElmAsString"> <#Include Label="MatElm"> <#Include Label="Cancel"> <#Include Label="LocalizeAt"> <#Include Label="LocalizeAtZero"> <#Include Label="LocalizePolynomialRingAtZeroWithMora"> <#Include Label="HomalgLocalRingElement"> <#Include Label="HomalgLocalMatrix">