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Module mathematics - Motivation | | Module mathematics - Motivation: Encyclopedia II - Module mathematics - Motivation | | In a vector space, the set of scalars forms a field and acts on the vectors by scalar multiplication, subject to certain formal laws such as the distributive law. In a module, the scalars need only be a ring, so the module concept represents a significant generalization.
Much of the theory of modules consists of extending as many as possible of the desirable properties of vector spaces to the realm of modules over a "well-behaved" ring, such as a principal ideal domain. However, modules can be quite a bit more complicate ...
See also:Module mathematics, Module mathematics - Motivation, Module mathematics - Definition, Module mathematics - Examples, Module mathematics - Submodules and homomorphisms, Module mathematics - Types of modules, Module mathematics - Relation to representation theory, Module mathematics - Generalizations | | | Module mathematics, Module mathematics - Definition, Module mathematics - Examples, Module mathematics - Generalizations, Module mathematics - Motivation, Module mathematics - Relation to representation theory, Module mathematics - Submodules and homomorphisms, Module mathematics - Types of modules, vector space, algebra (ring theory), module (model theory) | | |
| | | Module mathematics: Encyclopedia II - Module mathematics - Motivation
Module mathematics - Motivation
In a vector space, the set of scalars forms a field and acts on the vectors by scalar multiplication, subject to certain formal laws such as the distributive law. In a module, the scalars need only be a ring, so the module concept represents a significant generalization.
Much of the theory of modules consists of extending as many as possible of the desirable properties of vector spaces to the realm of modules over a "well-behaved" ring, such as a principal ideal domain. However, modules can be quite a bit more complicated than vector spaces; for instance, not all modules have a basis, and even those that do, free modules, behave significantly differently from vector spaces in some respects.
Modules form the core notion of commutative algebra, which is essential in many important fields of mathematics, including
- algebraic geometry,
- homological algebra and algebraic topology,
- representation theory of groups.
Other related archivesEuclidean space, Injective modules, Projective modules, abelian category, abelian group, abstract algebra, additive functor, algebra (ring theory), algebraic geometry, algebraic topology, annihilator, artinian module, basis, bijective, bimodule, cartesian product, category, commutative, commutative algebra, differential forms, direct sum, direct summands, distributive law, faithful module, field, finitely generated, free, free module, free modules, free product, functions, functor category, graded module, group endomorphism, groups, homological algebra, homomorphism, indecomposable module, injective, integers, isomorphism, isomorphism theorems, kernel, left ideal, linear combination, map, matrices, matrix multiplication, modular arithmetic, module (model theory), natural number, noetherian module, preadditive category, principal ideal domain, rank, real numbers, representation theory, ring, ring homomorphism, ringed space, scalars, set, sheaves, simple module, smooth functions, smooth manifold, subgroup, tensor fields, unital, vector fields, vector space, well-behaved
 Adapted from the Wikipedia article "Motivation", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki |
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