Length of a module

In abstract algebra, the length of a module is a measure of the module's "size". It is defined to be the length of the longest chain of submodules and is a generalization of the concept of dimension for vector spaces. 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. 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.

Let M be a (left or right) module over some ring R. Given a chain of submodules of M of the form

we say that n is the length of the chain. The length of M is defined to be the largest length of any of its chains. If no such largest length exists, we say that M has infinite length.

A ring R is said to have finite length as a ring if it has finite length as left R module.

The zero module is the only one with length 0. Modules with length 1 are precisely the simple modules.

For every finite-dimensional vector space (viewed as a module over the base field), the length and the dimension coincide.

The length of the cyclic group Z/nZ (viewed as a module over the integers Z) is equal to the number of prime factors of n, with multiple prime factors counted multiple times.

A module M has finite length if and only if it is both Artinian and Noetherian. (cf. Hopkins' theorem)

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