Modules

Chapter 9 initiates module theory, which is one of the most important topics in modern algebra. It is a generalization of an abelian group (which is a module over Z) and also a natural generalization of a vector space (which is a module over a division ring (field)). Many results of vector spaces are generalized in some special classes of modules, such as free modules and finitely generated modules over principal ideal domains. Modules are closely related to the representation theory of groups. One of the basic concepts which accelerates the study of commutative algebra is the module theory, as modules play the central role in commutative algebra. Modules are also widely used in structure theory of rings, additive abelian groups, homological algebra, algebraic geometry and algebraic topology. In this chapter the basic properties of modules are proved. Moreover, modules of special classes such as free modules, modules over principal ideal domains along with structure theorems, exact sequences of modules and their homomorphisms, Noetherian and Artinian modules, homology, and cohomology modules are studied. The study culminates in a discussion of the topology of the spectrum of modules and rings with special reference to Zariski topology.