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.

Definition

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

$N_0\subsetneq N_1 \subsetneq \cdots \subsetneq N_n$

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.

Examples

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.

Facts

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

If M has finite length and N is a submodule of M, then N has finite length as well, and we have length(N) ≤ length(M). Furthermore, if N is a proper submodule of M (i.e. if it is unequal to M), then length(N) < length(M).

If the modules M₁ and M₂ have finite length, then so does their direct sum, and the length of the direct sum equals the sum of the lengths of M₁ and M₂.

Suppose

$0\rarr L \rarr M \rarr N \rarr 0$

is a short exact sequence of R-modules. Then M has finite length if and only if L and N have finite length, and we have

length(M) = length(L) + length(N).

(This statement implies the two previous ones.)

A composition series of the module M is a chain of the form

$0=N_0\subsetneq N_1 \subsetneq \cdots \subsetneq N_n=M$

such that

$N_{i+1}/N_i \mbox{ is simple for }i=0,\dots,n-1$

Every finite-length module M has a composition series, and the length of every such composition series is equal to the length of M.

See also

• Hilbert–Poincaré series

References

• Steven H. Weintraub, Representation Theory of Finite Groups AMS (2003) ISBN 0-8218-3222-0, ISBN 978-0-8218-3222-6