Modules are to rings as vector spaces are to fields.
Modules are to rings as sets with group actions are to groups.
Definition of (left) modules
Definition: Let be a ring (for now, not necessarily commutative and not necessarily having a unit). A left -module is an abelian group together with a map (written ) such that:
If has a unit element , we also require for all .
Right modules
A right module is defined by a map and written and satisfying the property
If is not commutative, these really are different, since for a left module:
acts by “first , then
while for a right module
acts by “first , then .”
Left and Right modules
If is commutative, and is a left -module, then we can define a right module with the same underlying abelian group and by defining . This works because
Remarks
Vector spaces
If is a field, then a left (or right) -module is the same as a vector space.
Another definition
If is an abelian group, and is a ring, then a left -module structure on is the same as a ring map
If is the endomorphism associated to , then . The associativity comes from defining the ring structure on as the usual composition of functions:
Submodules
Definition: If is a left -module, then a submodule of is a subgroup with the property that, if , then for all .
Observation: A ring is a left module over itself by ring multiplication. The (left) ideals of are exactly the left submodules of .
Essential examples
Rings as modules over themselves
Every ring is a left module over itself. The submodules of are the left ideals.
is also a right module over itself, with the right ideals being the right submodules.
If is a field and , let be the matrix ring over . The matrices with arbitrary first column and zeros elsewhere form a left ideal and therefore a left submodule of as left -module. But is not a right -submodule.
A field is a one-dimensional vector space over itself, and a commutative ring is a module (left and right) over itself with the ideals of being the submodules.
Free modules
Let be a ring with unity and let be a positive integer. Then
is an module with componentwise addition and multiplication given by .
This is called the free -module of rank .
Free modules and vector spaces
If is a field, the free -module of rank is an -dimensional vector space.
The submodules of a finite dimensional vector space are all subspaces which are copies of for .
For more general the picture is more complicated. Let and . Then:
is a submodule of which “looks like” a subspace.
is a submodule of which does not.
Change of rings (restriction of scalars)
An abelian group may be an module for different rings . For example:
is a module over , where it is a one dimensional vector space and its only -submodules are and itself.
is a module over , and it has many -submodules, such as .
More generally, if is a subring, and is an -module, then it is an -module. This is called restriction of scalars.
-modules are the same as abelian groups
Let be an abelian group. Then it is automatically a -module where we define
Furthermore, given any -module, it must be the case that
(Note: this is why we require when is a ring with unity in the module axioms).
Further, submodules of (as -module) are just the subgroups of (as abelian group).
Change of rings (quotients)
Suppose that is a left module and is a two-sided ideal with the property that, for all , and all , we have . In this case we say that annihilates or that .
With this hypothesis, we may view as an module by defining for any coset representative . This is well-defined since two different coset representatives satisfy for some and therefore since .
If is an abelian group and is a positive integer such that , then can be viewed as a module over by this process.
This operation is a special case of a general operation called base change or extension of scalars that we will study in more detail later.
Modules over
Basic construction
Let be a field, let be a vector space over , and let be an -linear transformation. Define a homomorphism
by sending
This construction makes into a module for which depends on the choice of the linear transformation .
Polynomials and linear transformations
Let and let be the linear transformation given by the matrix
If and are the standard basis elements of then
Polynomials and linear transformations continued
Therefore and
so the polynomial is in the kernel of the map from .
By the base change construction above this means that can be viewed as a module over .
Characterization of modules
We saw above that, given an -vector space with a linear transformation , we get an module where acts on through .
Conversely, suppose that is an module over . Then is an vector space (via the restriction of scalars from to ). Furthermore, the element acts on as an -linear transformation because that’s what the module axioms amount to.
Therefore there is an equivalence between
Submodules of modules
In the correspondence above, a submodule of an module corresponds to a subspace that is preserved by , meaning .
Thus, not all subspaces of correspond to submodules.
In the example given earlier, the only -stable proper subspace of is the zero subspace.
If we consider instead the linear map on satisfying and , then the one dimensional subspace spanned by is -stable and viewed as an module via has a submodule corresponding to that subspace.
Checking the submodule property
Proposition: A subset of a left -module is a submodule if it is nonempty and, for all and , we have . Alternatively, if is a subgroup of the abelian group and for all then is a submodule.
Algebras
Definition: Let be a commutative ring with unity. An -algebra is a (not necessarily commutative) ring with a ring homomorphism carrying to such that is in the center of .
The polynomial ring is an -algebra, as is the matrix ring where the homomorphism embeds as the diagonal matrices. More generally, any -algebra , where is a field, contains in its center and the identites of and are the same.
The ring is a -algebra. In fact any ring with is a algebra by the map sending to .
The ring is a algebra.
We typically omit the explicit map and just think of as “contained in” ; this can be misleading since doesn’t need to be injective, but it works in practice.
Algebra morphisms
Definition: A map of -algebras is a ring homomorphism that is -linear in the sense that for all and .
Any homomorphism of rings with unity is a -algebra morphism.
Modules Homomorphisms, Quotient Modules, and Mapping Properties
Module homomorphisms
Definition: Let be a ring and let and be (left) -modules. A function is an -module homomorphism if:
it is a homomorphism between the abelian group structures on and
it is -linear, meaning for all .
Note that, if is a field, then and are vector spaces and an -module homomorphism is just a linear map.
A module isomorphism is a bijective homomorphism.
We let denote the set of -module homomorphisms from to .
Kernels and images
Let be a ring and let and be -modules. Let be a homomorphism.
Let (the kernel of ). This is a submodule of .
Let be the image of . Then is a submodule of .
Quotient modules
Let be an module and let be a submodule.
Definition: Let be the quotient abelian group. Then is an -module where acts on cosets by
This is called the quotient module of by .
The -module structure is well defined because if , then for some , and . Since is a submodule, so .
Notice that can be any submodule, there is no “normality” condition like for groups.
There is always a “projection” homomorphism defined by which has kernel .
Sums of modules
If and are submodules of a module , then is the smallest submodule of containing both and . Alternatively it is:
Mapping Properties
Let , , and be modules, and let be a homomorphism with . Then there is a unique homomorphism making this diagram commutative:
Isomorphism theorems
The isomorphism theorems for abelian groups give isomorphism theorems for modules.
If is a homomorphism, then the map gives an isomorphism between and .
is isomorphic to .
is isomorphic to .
There is a bijection between the lattice of submodules of and submodules of containing given by .
The proofs of all of these facts are found by checking that the group isomorphisms respect the action of the ring .
The set is an abelian group: and the zero map is the identity.
If is commutative then is an -module if we set to be the function . We need to be a module homomorphism, which means we need:
This works out ok if is commutative since
but it fails if is not commutative.
The set is a ring with multiplication given by composition. The identity map gives an identity for this ring.
If is commutative then, given , we have an element given by . This is a homomorphism because
but this fails in general if is not commutative. Thus, if is commutative, is an -algebra.
More on
If , then is the ring of matrices with entries from .