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. Definition Length of a module Let be a (left or right) module over some ring . Given a chain of submodules of of the form we say that is the length of the chain. The length of is defined to be the largest length of any of its chains. If no such largest length exists, we say that has infinite length. Length of a ring A ring is said to have finite length as a ring if it has finite length as a left -module. Properties Finite length and finite modules If an -module has finite length, then it is finitely generated. If R is a field, then the converse is also true. Relation to Artinian and Noetherian modules An -module has finite length if and only if it is both a Noetherian module and an Artinian module (cf. Hopkins' theorem). Since all Artinian rings are Noetherian, this implies that a ring has finite length if and only if it is Artinian. Behavior with respect to short exact sequences Supposeis a short exact sequence of -modules. Then M has finite length if and only if L and N have finite length, and we have In particular, it implies the following two properties The direct sum of two modules of finite length has finite length The submodule of a module with finite length has finite length, and its length is less than or equal to its parent module. Jordan–Hölder theorem A composition series of the module M is a chain of the form such that A module M has finite