Introduction and definition
MotivationIn a vector space, the set of scalars is a and acts on the vectors by scalar multiplication, subject to certain axioms such as the . In a module, the scalars need only be a ring, so the module concept represents a significant generalization. In commutative algebra, both and s 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 " " ring, such as a . However, modules can be quite a bit more complicated than vector spaces; for instance, not all modules have a , and even those that do, s, need not have a unique if the underlying ring does not satisfy the condition, unlike vector spaces, which always have a (possibly infinite) basis whose cardinality is then unique. (These last two assertions require the in general, but not in the case of finite-dimensional spaces, or certain well-behaved infinite-dimensional spaces such as L''p'' spaces.)
Formal definitionSuppose that ''R'' is a ring, and 1 is its multiplicative identity. A left ''R''-module ''M'' consists of an and an operation such that for all ''r'', ''s'' in ''R'' and ''x'', ''y'' in ''M'', we have # # # # 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 , all rings and modules are assumed to be unital. An ''(R,S)''- 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 for all ''r'' in ''R'', ''x'' in ''M'', and ''s'' in ''S''. If ''R'' is , then left ''R''-modules are the same as right ''R''-modules and are simply called ''R''-modules.
Examples*If ''K'' is a , then ''K''- s (vector spaces over ''K'') and ''K''-modules are identical. *If ''K'' is a field, and ''K'' 'x''a univariate , then a module.html" ;"title="'x''
Submodules and homomorphismsSuppose ''M'' is a left ''R''-module and ''N'' is a of ''M''. Then ''N'' is a submodule (or more explicitly an ''R''-submodule) if for any ''n'' in ''N'' and any ''r'' in ''R'', the product (or for a right ''R''-module) is in ''N''. If ''X'' is any of an ''R''-module, then the submodule spanned by ''X'' is defined to be where ''N'' runs over the submodules of ''M'' which contain ''X'', or explicitly , which is important in the definition of tensor products. The set of submodules of a given module ''M'', together with the two binary operations + and ∩, forms a lattice which satisfies the modular law: Given submodules ''U'', ''N''1, ''N''2 of ''M'' such that , then the following two submodules are equal: . If ''M'' and ''N'' are left ''R''-modules, then a is a homomorphism of ''R''-modules if for any ''m'', ''n'' in ''M'' and ''r'', ''s'' in ''R'', :. This, like any of mathematical objects, is just a mapping which preserves the structure of the objects. Another name for a homomorphism of ''R''-modules is an ''R''- . A module homomorphism is called a module , and the two modules ''M'' and ''N'' are called isomorphic. Two isomorphic modules are identical for all practical purposes, differing solely in the notation for their elements. The of a module homomorphism is the submodule of ''M'' consisting of all elements that are sent to zero by ''f'', and the of ''f'' is the submodule of ''N'' consisting of values ''f''(''m'') for all elements ''m'' of ''M''. The s familiar from groups and vector spaces are also valid for ''R''-modules. Given a ring ''R'', the set of all left ''R''-modules together with their module homomorphisms forms an , denoted by ''R''-Mod (see ).
Types of modules; Finitely generated: An ''R''-module ''M'' is finitely generated if there exist finitely many elements ''x''1, ..., ''x''''n'' in ''M'' such that every element of ''M'' is a of those elements with coefficients from the ring ''R''. ; Cyclic: A module is called a if it is generated by one element. ; Free: A is a module that has a basis, or equivalently, one that is isomorphic to a of copies of the ring ''R''. These are the modules that behave very much like vector spaces. ; Projective: s are s of free modules and share many of their desirable properties. ; Injective: s are defined dually to projective modules. ; Flat: A module is called flat if taking the of it with any of ''R''-modules preserves exactness. ; Torsionless: A module is called torsionless if it embeds into its algebraic dual. ; Simple: A ''S'' is a module that is not and whose only submodules are and ''S''. Simple modules are sometimes called ''irreducible''.Jacobson (1964)
Relation to representation theoryA representation of a group ''G'' over a field ''k'' is a module over the ''k'' 'G'' If ''M'' is a left ''R''-module, then the ''action'' of an element ''r'' in ''R'' is defined to be the map that sends each ''x'' to ''rx'' (or ''xr'' in the case of a right module), and is necessarily a of the abelian group . The set of all group endomorphisms of ''M'' is denoted EndZ(''M'') and forms a ring under addition and , and sending a ring element ''r'' of ''R'' to its action actually defines a from ''R'' to EndZ(''M''). Such a ring homomorphism is called a ''representation'' of ''R'' over the abelian group ''M''; an alternative and equivalent way of defining left ''R''-modules is to say that a left ''R''-module is an abelian group ''M'' together with a representation of ''R'' over it. Such a representation may also be called a ''ring action'' of on . A representation is called ''faithful'' if and only if the map is . In terms of modules, this means that if ''r'' is an element of ''R'' such that for all ''x'' in ''M'', then . Every abelian group is a faithful module over the s or over some Z/''n''Z.
GeneralizationsA ring ''R'' corresponds to a R with a single . With this understanding, a left ''R''-module is just a covariant from R to the category Ab of abelian groups, and right ''R''-modules are contravariant additive functors. This suggests that, if C is any preadditive category, a covariant additive functor from C to Ab should be considered a generalized left module over C. These functors form a C-Mod which is the natural generalization of the module category ''R''-Mod. Modules over ''commutative'' rings can be generalized in a different direction: take a (''X'', O''X'') and consider the sheaves of O''X''-modules (see sheaf of modules). These form a category O''X''-Mod, and play an important role in modern . If ''X'' has only a single point, then this is a module category in the old sense over the commutative ring O''X''(''X''). One can also consider modules over a . Modules over rings are abelian groups, but modules over semirings are only s. Most applications of modules are still possible. In particular, for any ''S'', the matrices over ''S'' form a semiring over which the tuples of elements from ''S'' are a module (in this generalized sense only). This allows a further generalization of the concept of incorporating the semirings from theoretical computer science. Over near-rings, one can consider near-ring modules, a nonabelian generalization of modules.
See also* * * Module (model theory) * Module spectrum * Annihilator
References* F.W. Anderson and K.R. Fuller: ''Rings and Categories of Modules'', Graduate Texts in Mathematics, Vol. 13, 2nd Ed., Springer-Verlag, New York, 1992, , * . ''Structure of rings''. Colloquium publications, Vol. 37, 2nd Ed., AMS Bookstore, 1964,