Table of Contents
Vector Space over a Field
Vector spaces over a field are a special case of the more general notion of modules over a ring. Normally, textbooks define a vector space as a set equipped with two operations which obey a long list of axioms:
An (abstract) vector space consists of
- A field , whose elements are called scalars
- A set , whose elements are called (abstract) vectors
- A rule for vector addition, satisfying
- (additive identity) exists with for all
- (additive inverse) for all , exists with
- A rule for scalar multiplication, satisfying
- (scalar identity) for all
- (distributes over vector addition)
- (distributes over field addition)
We can state these properties more concisely by noticing that Property III is equivalent to the requirement that forms a commutative group.
An (abstract) vector space over the field is a commutative group together with a rule satisfying
- (scalar identity) for all
- (distributes over addition)
- (distributes over field addition)
Definitions 1a and 1b seem to present the set as the primary object of interest, relegating the scalars to the sidelines. The key to understanding modules is to turn this presumption on its head by treating as the distinguished object instead.
By partial application of the scaling operator , each scalar corresponds to a linear map from to itself. Linear self-maps on constitute the endomorphism ring , whose operations are pointwise addition and function composition. The vector space axioms ensure that the map from field elements to linear self-maps is a ring homomorphism. We arrive at our third and final definition,
An (abstract) vector space over the field is a commutative group together with a ring homomorphism .
The ring homomorphism defines the additive and multiplicative group actions on by scalars from the field .
Module over a Ring
For modules, we require only that the set acting on be a ring, rather than a field.
A module over the ring is a commutative group together with a ring homomorphism defining an action of on , where is the set of group homomorphisms .
Modules over a ring are called -modules, for short. An -module is called left if it arises from a left action, and right otherwise. As for vector spaces, we could unfold this definition into a list of axioms, but this would obfuscate the real purpose of modules: Many mathematical objects happen to be rings, and modules allow us to study rings by their action on a set (much like we can study groups via their representations).
Let be an -module. An -submodule of is a subgroup closed under the ring action, for , .
Several important examples of modules are listed below.
- If is a field, then -modules and -vector spaces are identical.
- Every ring is an -module over itself. In particular, every field is an -vector space. Submodules of as a field over itself are ideals.
- If is a subring of with , every -module is an -module.
- If is a commutative group of finite order , then for all , and is a -module. In particular, if has prime order , then is a vector space over the field .
- The smooth real-valued functions on a smooth manifold form a ring. The smooth vector fields on form a -module.
- For a ring , every -algebra has natural (left/right) -module structure given by the (left/right) ring action of on .
(-modules) By definition, every -module is a commutative group. Likewise, every commutative group becomes a -module under the ring action defined for , by We conclude that -modules and commutative groups are one in the same.
Modules over a Polynomial Ring
The polynomial ring is the space of formal linear combinations of powers of an indeterminate , with coefficients drawn from an underlying field .
Polynomials form a ring under entrywise addition and discrete convolution of coefficient sequences. The sum and product of have coefficients
Consider what it would mean for an -vector space to be an -module. We need a ring homomorphism describing the action of polynomials on vectors. Since preserves sums and products between and as rings, we find that the choice of a single linear map determines the value of on arbitrary polynomials~, Similarly, any choice of yields a valid ring homomorphism, exposing a bijection between -modules and pairs . In general, there are many different -module structures a given -vector space , each corresponding to a choice of linear .
The -submodules of an -module~ are precisely the -invariant subspaces of , where denotes the action of .
Each -submodule of is closed under actions by ring elements, including . Likewise, every -invariant subspace is closed under ring actions, which are all polynomials in .
An -module homomorphism is a map between modules which respects the -module structure, by preserving addition and commuting with the ring action on ,
The kernel of a module homomorphism is its kernel as an additive group homomorphism. A bijective -module homomorphism is an isomorphism. For any ring , the set of homomorphisms between two -modules forms a commutative group under pointwise addition, for . Moreover,
For a commutative ring , the group forms an -module under the ring action given by
Commutativity of guarantees that , since
Ring of Module Endomorphisms
Endomorphisms form a unital ring, where We write for the endomorphism ring of .
Let be a module over a commutative ring . The endomorphism ring forms an -algebra, under the same ring action which defines as an -module.
This property is normally stated without reference to ring homomorphisms, but in these notes we wish to emphasize that the study of modules is really the study of . There is at least one subtlety, though: When defining as an -module, we required that be a ring homomorphism from to the additive group endomorphisms on . Now, we are asking whether each is also an -module homomorphism.
First, the additive group homomorphism is also a module homomorphism, since for and , Futher, sending is a ring homomorphism. Finally, each commutes with every element ,
By definition, every field is a commutative ring. Therefore, the endomorphisms of any -vector space form an -algebra.
For groups and rings, recall that quotients are well-defined only for subgroups and subrings (ideals), respectively. For modules , it turns out that submodule has a quotient , and the natural projection map is a ring homomorphism with kernel . Similarly, each -vector subspace has a quotient -vector space arising as the kernel of some linear map.
Let be a ring. Let be a submodule of the -module . The (additive, commutative) quotient group can be made into an -module under the ring action given by The natural projection mapping is an -module homomorphism with kernel .
(First Isomorphism Theorem) Let be -modules. The kernel of any module homomorphism is a submodule of , and
The vector space concepts of linear combinations, bases, and span all have analogues in -module theory. We normally assume is a ring with identity.
Let be an -module. The submodule of generated by a subset is the set of finite -linear combinations A submodule is finitely generated if is finite. A cyclic submodule is generated by a single element .
An -module is free on the subset if each nonzero expands uniquely as an -linear combination of elements from , in which case is called a basis for .
In general, more than one basis may exist. If is commutative, every basis has the same cardinality, called the module rank of . Unlike for vector spaces, not every module has a basis (not every module is free).
Universal Property of Free Modules
Recall that every linear map between -vector spaces is uniquely determined by its value on points. -linear maps between free modules enjoy the same property, which is normally stated in the following way: