Localizations Of Poisson Algebras And Lie-Rinehart Algebras

Localization is a fundamental method for the study of commutative al-gebras and algebraic geometries.Recently,Poisson algebras and Lie-Rinehart algebra were widely investigated by many mathematicians.As we know,these two algebraic object are originated from geometry,such as Poisson manifolds,Poisson varieties and the smooth function ring and tangent vector fields on a smooth manifold.In order to consider the local properties of these geometric objects,it is natural to study the localization theory for the corresponding algebraic objects.In the first chapter,we reviewed the development history and background of localization,Poisson algebra and their modules and Lie-Rinehart algebra,and recall the localization theory for commutative algebras and their modules.In the second chapter,we firstly recall some definitions of Poisson alge-bras and Poisson module.By a multiplication closed set,we introduce the localization theory of Poisson algebras and Poisson modules respectively.Finally,we recall the original definition and its equivalent definition of Lie-Rinehart algebra.Considering a multiplication closed set,we introduce the localization theory of Lie-Rinehart algebra.