In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the "scalars" may lie in an arbitrary ring.
Thus, a module, like a vector space, is an additive 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) and distributive.
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.
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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. In commutative algebra, it is important that 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 important ring theoretic conditions can be expressed either about left ideals or left modules.
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 modules, 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 basis whose cardinality is then unique (assuming the axiom of choice).
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
If one writes the scalar action as fr so that fr(x) = rx, and f for the map which takes each r to its corresponding map fr, then the first axiom states that every fr 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 (cf. 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 RM. A right R-module M or MR 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 above in the definition of an R-module, and so 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.)
A bimodule is a module which is a left module and a right module such that the two multiplications are compatible.
If R is commutative, then left R-modules are the same as right R-modules and are simply called R-modules.
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).
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, N1, N2 of M such that N1 ⊂ N2, then the two submodules are equal: (N1 + U) ∩ N2 = N1 + (U ∩ N2).
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,
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 kernel of a module homomorphism f : M → N is the submodule of M consisting of all elements that are sent to zero by f. The isomorphism theorems familiar from 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.
Finitely generated. A module M is finitely generated if there exist finitely many elements x1,...,xn 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 a 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 modules are direct summands of free modules and share many of their desirable properties.
Injective. Injective modules 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 such that 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 is a module decomposable as a direct sum M = ⊕x Mx over a graded ring R = ⊕x Rx such that RxMy ⊂ Mx + y for all x and y.
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/nZ.
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, OX) and consider the sheaves of OX-modules. These form a category OX-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 OX(X).
One can also consider modules over a semiring. Modules over rings are abelian groups, but modules over semirings are only commutative monoids. Most applications of modules are still possible. In particular, for any semiring 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 vector space incorporating the semirings from theoretical computer science.