submodule

In mathematics, a module is a generalization of the notion of vector space, wherein the field of scalars is replaced by a ring. The concept of ''module'' is also a generalization of the one of abelian group, since the abelian groups are exactly the modules over the ring of integers.

Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operation of addition between elements of the ring or module and is compatible with the ring multiplication.

Modules are very closely related to the representation theory of groups. They are also one of the central notions of commutative algebra and homological algebra, and are used widely in

algebraic geometry

and algebraic topology.

Introduction and definition

Motivation

In a vector space, the set of scalars is a field and acts on the vectors by scalar multiplication, subject to certain axioms such as the distributive law. In a module, the scalars need only be a ring, so the module concept represents a significant generalization. In commutative algebra, both ideals and quotient rings are modules, so that many arguments about ideals or quotient rings can be combined into a single argument about modules. In non-commutative algebra, the distinction between left ideals, ideals, and modules becomes more pronounced, though some ring-theoretic conditions can be expressed either about left ideals or left modules.

Much of the theory of modules consists of extending as many of the desirable properties of vector spaces as possible to the realm of modules over a "well-behaved" ring, such as a principal ideal domain. However, modules can be quite a bit more complicated than vector spaces; for instance, not all modules have a basis, and even those that do,

free module

s, need not have a unique

rank

if the underlying ring does not satisfy the invariant basis number condition, unlike vector spaces, which always have a (possibly infinite) basis whose cardinality is then unique. (These last two assertions require the

axiom of choice

in general, but not in the case of finite-dimensional spaces, or certain well-behaved infinite-dimensional spaces such as L^''p'' spaces.)

Formal definition

Suppose that ''R'' is a ring, and 1 is its multiplicative identity.
A left ''R''-module ''M'' consists of an abelian group and an operation such that for all ''r'', ''s'' in ''R'' and ''x'', ''y'' in ''M'', we have
#$r \cdot ( x + y ) = r \cdot x + r \cdot y$
#$( r + s ) \cdot x = r \cdot x + s \cdot x$
#$( r s ) \cdot x = r \cdot ( s \cdot x )$
#$1 \cdot x = x .$

The operation ⋅ is called ''scalar multiplication''. Often the symbol ⋅ is omitted, but in this article we use it and reserve juxtaposition for multiplication in ''R''. One may write _''R''''M'' to emphasize that ''M'' is a left ''R''-module. A right ''R''-module ''M''_''R'' is defined similarly in terms of an operation .

Authors who do not require rings to be unital omit condition 4 in the definition above; they would call the structures defined above "unital left ''R''-modules". In this article, consistent with the glossary of ring theory, all rings and modules are assumed to be unital.

An ''(R,S)''-bimodule is an abelian group together with both a left scalar multiplication ⋅ by elements of ''R'' and a right scalar multiplication * by elements of ''S'', making it simultaneously a left ''R''-module and a right ''S''-module, satisfying the additional condition $(r \cdot x) \ast s = r \cdot ( x \ast s )$ for all ''r'' in ''R'', ''x'' in ''M'', and ''s'' in ''S''.

If ''R'' is commutative, then left ''R''-modules are the same as right ''R''-modules and are simply called ''R''-modules.

Examples

*If ''K'' is a field, then ''K''-vector spaces (vector spaces over ''K'') and ''K''-modules are identical.
*If ''K'' is a field, and ''K'' 'x''a univariate polynomial ring, then a

''K''

module.html" ;"title="'x''