Length of a module

# Length of a module

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In abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

, the length of a module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...

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 space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

s. 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
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...

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
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...

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.

## Examples

The zero module is the only one with length 0. Modules with length 1 are precisely the simple module
Simple module
In mathematics, specifically in ring theory, the simple modules over a ring R are the modules over R which have no non-zero proper submodules. Equivalently, a module M is simple if and only if every cyclic submodule generated by a non-zero element of M equals M...

s.

For every finite-dimensional vector space (viewed as a module over the base field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

), the length and the dimension coincide.

The length of the cyclic group
Cyclic group
In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...

Z/nZ (viewed as a module over the integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

s Z)
is equal to the number of prime
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

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
Artinian module
In abstract algebra, an Artinian module is a module that satisfies the descending chain condition on its poset of submodules. They are for modules what Artinian rings are for rings, and a ring is Artinian if and only if it is an Artinian module over itself...

and Noetherian
Noetherian module
In abstract algebra, an Noetherian module is a module that satisfies the ascending chain condition on its submodules, where the submodules are partially ordered by inclusion....

.

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
Direct sum of modules
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The result of the direct summation of modules is the "smallest general" module which contains the given modules as submodules...

, and the length of the direct sum equals the sum of the lengths of M1 and M2.

Suppose
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
Composition series
In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many naturally occurring modules are not semisimple, hence...

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