In abstract algebra, the length of a module is a measure of the module's "size". It is defined as the length of the longest ascending chain of submodules and is a generalization of the concept of dimension for vector spaces. The modules with finite length share many important properties with finite-dimensional vector spaces. Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... In abstract algebra, the notion of a module over a ring is the common generalizations of two of the most important notions in algebra, vector space (where we take the ring to be a particular field), and abelian group (where we take the ring to be the ring of integers). ... In abstract algebra, a module is a generalization of a vector space. ... In mathematics, the dimension of a vector space V is the cardinality (i. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...

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. In commutative algebra, if R is a commutative ring and M an R-module, a R-regular sequence on M is a d-tuple of (non-zero non-unit) elements r1, r2, ..., rd from R such that for each i, ri is not a zerodivisor on the quotient R-module... In commutative algebra, the height of an ideal I in a ring R is the number of strict inclusions in the longest chain of prime ideals contained in I. In a Noetherian ring, Krulls height theorem says that the height of an ideal generated by n elements is no... In common usage, the dimensions (from Latin measured out) of an object are the parameters or measurements required to define its shape and size, that is, usually, its height, width, and length. ...

Definition

Let M be a (left or right) module over some ring R. Given a chain of submodules of M of the form In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ...

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.

Examples

The zero module is the only one with length 0. Modules with length 1 are precisely the simple modules. In abstract algebra, a (left or right) module S over a ring R is called simple if it is not the zero module and if its only submodules are 0 and S. Understanding the simple modules over a ring is usually helpful because they form the building blocks of all...

For every finite-dimensional vector space (viewed as a module over the base field), the length and the dimension coincide. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

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. 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 a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. ...

Facts

A module M has finite length if and only if it is both Artinian and Noetherian. In abstract algebra, an Artinian module is a module that satisfies the descending chain condition on its submodules. ... In ring theory, if R is a ring and M is a module over R, then M is Noetherian if M satisfies the ascending chain condition on its submodules when they are 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, and the length of the direct sum equals the sum of the lengths of M₁ and M₂. In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ...

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 In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ...

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 In mathematics, a composition series of a group G is a chain of subgroups of G satisfying where stands for normal subgroup, such that each quotient group Hi+1/Hi is a simple group. ...

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.