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Thus, a module, like a vector space, is an Abelian Group ; a product is defined between elements of the ring and elements of the module, and this multiplication is associative (when used with the multiplication in the ring), distributive. Modules are very closely related to the Representation Theory of Group s. They are also one of the central notions of Commutative Algebra and Homological Algebra , and are used widely in Algebraic Geometry and Algebraic Topology . MOTIVATION In a vector space, the set of Scalars forms a '' Field '' and acts on the vectors by scalar multiplication, subject to certain formal laws such as the Distributive Law . In a module, the scalars need only be a '' Ring '', so the module concept represents a significant generalization. Much of the theory of modules consists of extending as many as possible of the desirable properties of vector spaces 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, behave significantly differently from vector spaces in some respects. FOMAL DEFINITION A left ''R''-module over the ring ''R'' consists of an Abelian Group (''M'', +) and an operation ''R'' × ''M'' → ''M'' (called ''scalar multiplication'', usually just written by juxtaposition, i.e. as ''rx'' for ''r'' in ''R'' and ''x'' in ''M'') such that For all ''r'',''s'' in ''R'', ''x'',''y'' in ''M'', we have # ''r''(''x''+''y'') = ''rx''+''ry'' # (''r''+''s'')''x'' = ''rx''+''sx'' # (''rs'')''x'' = ''r''(''sx'') # 1''x'' = ''x'' If one writes the scalar action as ''f''''r'' so that ''f''''r''(''x'') = ''rx'', and ''f'' for the map which takes each ''r'' to its corresponding map ''f''''r'', then the first axiom states that every ''f''''r'' is a Group Homomorphism of ''M'', and the other three axioms assert that ''f'' is a Ring Homomorphism from ''R'' to the Endomorphism Ring End(''M''). Thus a module is a ring action on an abelian group (confer Group Action ). In this sense, module theory generalizes Representation Theory , which deals with group actions on vector spaces, or equivalently group ring actions. Usually, we simply write "a left ''R''-module ''M''" or ''R''''M''. A right ''R''-module ''M'' or ''M''''R'' is defined similarly, only the ring acts on the right, i.e. we have a scalar multiplication of the form ''M'' × ''R'' → ''M'', and the above axioms are written with scalars ''r'' and ''s'' on the right of ''x'' and ''y''. Authors who do not require rings to be Unital omit condition 4 in the above definition, and call the above structures "unital left modules". In this article however, all rings and modules are assumed to be unital. A Bimodule is a module which is both a left module and a right module. If ''R'' is Commutative , then left ''R''-modules are the same as right ''R''-modules and are simply called ''R''-modules. EXAMPLES
SUBMODULES AND HOMOMORPHISMS Suppose ''M'' is a left ''R''-module and ''N'' is a Subgroup of ''M''. Then ''N'' is a submodule (or ''R''-submodule, to be more explicit) if, for any ''n'' in ''N'' and any ''r'' in ''R'', the product ''rn'' is in ''N'' (or ''nr'' for a right module). If ''M'' and ''N'' are left ''R''-modules, then a Map ''f'' : ''M'' → ''N'' is a homomorphism of R-modules if, for any ''m, n'' in ''M'' and ''r, s'' in ''R'', f This, like any Homomorphism of mathematical objects, is just a mapping which preserves the structure of the objects. A Bijective module homomorphism is an Isomorphism of modules, and the two modules are called ''isomorphic''. Two isomorphic modules are identical for all practical purposes, differing solely in the notation for their elements. The s familiar from abelian groups and vector spaces are also valid for ''R''-modules. The left ''R''-modules, together with their module homomorphisms, form a Category , written as ''R''-Mod. This is an Abelian Category . TYPES OF MODULES Finitely generated. A 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 Linear Combination of those elements with coefficients from the scalar ring ''R''. Cyclic module. A module is called Cyclic Module if it is generated by one element. Free. A Free Module is a module that has a basis, or equivalently, one that is isomorphic to a Direct Sum of copies of the scalar ring ''R''. These are the modules that behave very much like vector spaces. Projective. Projective Module s are Direct Summand s of free modules and share many of their desirable properties. Injective. Injective Module s are defined dually to projective modules. Simple. A Simple Module ''S'' is a module that is not {0} and whose only submodules are {0} and ''S''. Simple modules are sometimes called ''irreducible''. Indecomposable. An Indecomposable Module is a non-zero module that cannot be written as a Direct Sum of two non-zero submodules. Every simple module is indecomposable. Faithful. A Faithful Module ''M'' is one where the action of each ''r'' ≠ 0 in ''R'' on ''M'' is nontrivial (i.e. ''rx'' ≠ 0 for some ''x'' in ''M''). Equivalently, the Annihilator of ''M'' is the zero ideal. Noetherian. A Noetherian Module is a module whose every submodule is finitely generated. Equivalently, every increasing chain of submodules becomes stationary after finitely many steps. Artinian. An Artinian Module is a module in which every decreasing chain of submodules becomes stationary after finitely many steps. Graded. A Graded Module can be decomposed as a Free Product of modules. RELATION TO REPRESENTATION THEORY If ''M'' is a left ''R''-module, then the ''action'' of an element ''r'' in ''R'' is defined to be the map ''M'' → ''M'' that sends each ''x'' to ''rx'' (or ''xr'' in the case of a right module), and is necessarily a Group Endomorphism of the abelian group (''M'',+). The set of all group endomorphisms of ''M'' is denoted EndZ(''M'') and forms a ring under addition and composition, and sending a ring element ''r'' of ''R'' to its action actually defines a Ring Homomorphism from ''R'' to EndZ(''M''). Such a ring homomorphism ''R'' → EndZ(''M'') 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. A representation is called ''faithful'' if and only if the map ''R'' → EndZ(''M'') is Injective . In terms of modules, this means that if ''r'' is an element of ''R'' such that ''rx''=0 for all ''x'' in ''M'', then ''r''=0. Every abelian group is a faithful module over the Integers or over some Modular Arithmetic Z/''n''Z. GENERALIZATIONS Any ring ''R'' can be viewed as a Preadditive Category with a single object. With this understanding, a left ''R''-module is nothing but a (covariant) Additive Functor from ''R'' to the category Ab of abelian groups. 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 Functor Category ''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 Ringed Space (''X'', O''X'') and consider the Sheaves of O''X''-modules. These form a category O''X''-Mod, and play an important role in the Scheme -theoretic approach to Algebraic Geometry . If ''X'' has only a single point, then this is a module category in the old sense over the commutative ring O''X''(''X''). SEE ALSO REFERENCES
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