Module theory

Length of a module

In abstract algebra, the length of a module is a generalization of the dimension of a vector space which measures its size. page 153 In particular, as in the case of vector spaces, the only modules of finite length are finitely generated modules. It is defined to be the length of the longest chain of submodules. Modules with finite length share many important properties with finite-dimensional vector spaces. Other concepts used to 'count' in ring and module theory are depth and height; these are both somewhat more subtle to define. Moreover, their use is more aligned with dimension theory whereas length is used to analyze finite modules. There are also various ideas of dimension that are useful. Finite length commutative rings play an essential role in functorial treatments of formal algebraic geometry and deformation theory where Artin rings are used extensively. (Wikipedia).

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Bézout's theorem | Chinese remainder theorem | Rational function | Complex analysis | Vector space | Krull–Schmidt theorem | Serre's multiplicity conjectures | Hilbert–Poincaré series | Algebraic variety | Noetherian module | Projective space | Dimension | Weierstrass factorization theorem | Hypersurface | Composition series | Meromorphic function | Affine variety | Dimension theory (algebra) | Dimension (vector space) | Intersection theory | Unit (ring theory) | Integer | Simple module | Artinian module | Multiplicity (mathematics) | Codimension | Cyclic group | Ring (mathematics) | Ring theory | Prime number | Scheme (mathematics) | Hilbert scheme | Abstract algebra | Local ring | Finitely generated module | Module (mathematics) | Commutative ring