Moduleleer

Moduleleer is a term used in mathematics to denote the study of modules, algebraic structures that generalize vector spaces by allowing scalars to come from a ring instead of a field. The field of moduleleer examines how modules behave under maps, constructions, and decompositions, and how these structures relate to the rings over which they are defined.

Objects in moduleleer are modules over a ring R. One distinguishes left modules and right modules depending on whether scalars act on the left or the right. Submodules, quotient modules, and homomorphisms between modules are central constructs. The study often focuses on how these objects interact through operations such as direct sums, tensor products, and Hom functors.

Important concepts include exact sequences, kernels, images, projective and injective modules, and free modules. Projective modules generalize free modules and play a key role in lifting properties; injective modules absorb morphisms. The category of all modules over a fixed ring forms a rich abelian category, enabling the use of homological methods to analyze structure and relationships between modules.

Over a principal ideal domain, finitely generated modules admit a structure theorem that yields a decomposition into a direct sum of cyclic modules, providing concrete invariants such as invariant factors and elementary divisors. This result connects moduleleer to classical linear algebra over rings and informs classifications in more complex settings.

Moduleleer has applications across mathematics, including representation theory, algebraic topology, algebraic geometry, and number theory, as well as in coding theory where modules over rings model linear codes. Historically, the study grew from attempts to extend linear algebra to ring contexts and from the development of homological and categorical approaches that deepen the understanding of modules.