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Module (mathematics)
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a (not necessarily commutative) ring. The concept of a module also generalizes the notion of an 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 operations 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.
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 for those that do (free modules), the number of elements in a basis need not be the same for all bases (that is to say that they may 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 vector spaces, or certain well-behaved infinite-dimensional vector spaces such as Lp spaces.)
Suppose that R is a ring, and 1 is its multiplicative identity. A left R-module M consists of an abelian group (M, +) and an operation · : R × M → M 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 RM to emphasize that M is a left R-module. A right R-module MR is defined similarly in terms of an operation · : M × R → M.
The qualificative of left- or right-module does not depend on whether the scalars are written on the left or on the right, but on the property 3: if, in the above definition, the property 3 is replaced by
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Module (mathematics)
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a (not necessarily commutative) ring. The concept of a module also generalizes the notion of an 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 operations 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.
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 for those that do (free modules), the number of elements in a basis need not be the same for all bases (that is to say that they may 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 vector spaces, or certain well-behaved infinite-dimensional vector spaces such as Lp spaces.)
Suppose that R is a ring, and 1 is its multiplicative identity. A left R-module M consists of an abelian group (M, +) and an operation · : R × M → M 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 RM to emphasize that M is a left R-module. A right R-module MR is defined similarly in terms of an operation · : M × R → M.
The qualificative of left- or right-module does not depend on whether the scalars are written on the left or on the right, but on the property 3: if, in the above definition, the property 3 is replaced by