Length of a module - Citizendia

In abstract algebra, the length of a module is a measure of the module's "size". Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars 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. In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars In Mathematics, the dimension of a Vector space V is the cardinality (i In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added The 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. In commutative and homological algebra depth is an important invariant of rings and modules. In Commutative algebra, the height of an Prime ideal \mathfrak{p} in a ring R is the number of strict There are also various ideas of dimension that are useful. In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it

Definition

Let M be a (left or right) module over some ring R. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real Given a chain of submodules of M of the form

$N_0\sub N_1 \sub \cdots \sub N_n$

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.

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 or irreducible if it is not the Zero

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

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 or monogenous group is a group that can be generated by a single element in the sense that the group has an The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1

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 Abstract algebra, an Noetherian module is a module that satisfies the Ascending chain condition on its Submodules where the submodules are

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. The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction

Suppose

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

is a short exact sequence of R-modules. In Mathematics, especially in Homological algebra and other applications of Abelian category theory as well as in Differential geometry and Group 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

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

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. 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

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
network: | |