Dimension of a module

In commutative algebra , a branch of mathematics, modules are generalizations of vector spaces . Every vector space has a basis that determines its dimension ; in contrast, modules are generally not free and have no base. In commutative algebra there are several concepts that generalize the notion of dimension from vector spaces to modules.

This article is about commutative algebra. In particular, all rings under consideration are commutative and have a one element. Ring homomorphisms map single elements onto single elements. For more details, see Commutative Algebra .

Definitions

Dimension of a module

If a module is above a ring , its dimension is defined as the Krull dimension of the ring modulo the annulator of : ${\displaystyle M}$ ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle M}$

${\displaystyle \mathrm {dim} (M):=\mathrm {dim} (R/\mathrm {Ann} (M))}$

The similarity between the term dimension of a module and the term dimension of a vector space is only of a linguistic nature: as a module, every vector space has dimension 0, since a body has Krull dimension 0.

Length of a module

→ Main article : Length (algebra)

If there is a module, then a normal series is in a chain ${\displaystyle M}$ ${\displaystyle M}$

${\displaystyle M=M_{0}\supsetneq M_{1}\supsetneq \dots \supsetneq M_{n}=0}$

A normal series is called a composition series if

${\displaystyle M_{i}/M_{i+1}(0\leq i<n)}$

is a simple module . ( is a simple module if and are the only sub- modules of .) ${\displaystyle N}$ ${\displaystyle 0}$ ${\displaystyle N}$ ${\displaystyle N}$

${\displaystyle M}$is called of finite length if there is a bound for the lengths of all normal series. The maximum of the lengths is called the length of ${\displaystyle M}$ and becomes with

${\displaystyle \ell (M)}$

designated.

The set of Jordan Holder states that a module having a composition series, has a finite length and that any composition series are of equal length.

Mü of a module

If a finitely generated module is called the number of elements of a shortest generating system of . ${\displaystyle M}$ ${\displaystyle R}$ ${\displaystyle \mu (M)}$ ${\displaystyle M}$

Examples

Vector spaces

If is a -dimensional vector space, then is ${\displaystyle V}$ ${\displaystyle n}$

• ${\displaystyle \mathrm {dim} (V)=0}$(its dimension as a module )
• ${\displaystyle l(V)=n}$
• ${\displaystyle \mu (V)=n}$

Regular local rings

Is a local ring with maximal ideal , then if and regular if: ${\displaystyle R}$ ${\displaystyle m}$ ${\displaystyle R}$

${\displaystyle \mu (m)=\mathrm {dim} (R)}$

The following applies to all rings:

${\displaystyle \mu (m)\leq \mathrm {dim} (R)}$

See also

• Projective dimension
• depth

literature

• Ernst Kunz : Introduction to Commutative Algebra and Algebraic Geometry , Vieweg (1980), ISBN 3-528-07246-6
• Atiyah , Macdonald : Introduction to Commutative Algebra , Addison-Wesley (1969), ISBN 0-2010-0361-9
• Brüske, Ischebeck, Vogel: Commutative Algebra , Bibliographisches Institut (1989), ISBN 978-3411140411
• H. Matsumura, Commutative algebra 1980 ISBN 0-8053-7026-9 .