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Concept
Length of a module
Summary
In algebra, the length of a module is a generalization of the dimension of a vector space which measures its size. page 153 It is defined to be the length of the longest chain of submodules.
The modules of finite length are finitely generated modules, but as opposite to vector spaces, many finitely generated modules have an infinite length. Finitely generated modules of finite length are also called Artinian modules and are at the basis of the theory of Artinian rings.
For vector spaces, the length equals the dimension. This is not the case in commutative algebra and algebraic geometry, where a finite length may occur only when the dimension is zero.
The degree of an algebraic variety is the length of the ring associated to the algebraic set of dimension zero resulting from the intersection of the variety with generic hyperplanes. In algebraic geometry, the intersection multiplicity is commonly defined as the length of a specific module.
Let be a (left or right) module over some ring . Given a chain of submodules of of the form
one says that is the length of the chain. The length of is the largest length of any of its chains. If no such largest length exists, we say that has infinite length. Clearly, if the length of a chain equals the length of the module, one has and
A ring is said to have finite length as a ring if it has finite length as a left -module.
If an -module has finite length, then it is finitely generated. If R is a field, then the converse is also true.
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.
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.
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