In
algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
, given a
ring ''R'', the category of left modules over ''R'' is the
category whose
objects are all left
modules over ''R'' and whose
morphisms are all
module homomorphisms between left ''R''-modules. For example, when ''R'' is the ring of
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s Z, it is the same thing as the
category of abelian groups
In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in Ab.
Properties
The zero object o ...
. The category of right modules is defined in a similar way.
One can also define the category of
bimodule
In abstract algebra, a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible. Besides appearing naturally in many parts of mathematics, bimodules play a clarifying role, i ...
s over a ring ''R'' but that category is equivalent to the category of left (or right) modules over the
enveloping algebra of ''R'' (or over the opposite of that).
Note: Some authors use the term module category for the category of modules. This term can be ambiguous since it could also refer to a category with a
monoidal-category action.
Properties
The categories of left and right modules are
abelian categories. These categories have
enough projectives and
enough injectives.
Mitchell's embedding theorem states every abelian category arises as a
full subcategory
In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitivel ...
of the category of modules over some ring.
Projective limits and
inductive limits exist in the categories of left and right modules.
Over a
commutative ring
In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
, together with the
tensor product of modules ⊗, the category of modules is a
symmetric monoidal category.
Objects
A
monoid object of the category of modules over a commutative ring ''R'' is exactly an
associative algebra over ''R''.
A
compact object in ''R''-Mod is exactly a finitely presented module.
Category of vector spaces
The
category ''K''-Vect (some authors use Vect
''K'') has all
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
s over a
field ''K'' as objects, and
''K''-linear maps as morphisms. Since vector spaces over ''K'' (as a field) are the same thing as
modules over the
ring ''K'', ''K''-Vect is a special case of ''R''-Mod (some authors use Mod
''R''), the category of left ''R''-modules.
Much of
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrix (mathemat ...
concerns the description of ''K''-Vect. For example, the
dimension theorem for vector spaces
In mathematics, the dimension theorem for vector spaces states that all bases of a vector space have equally many elements. This number of elements may be finite or infinite (in the latter case, it is a cardinal number), and defines the dimension ...
says that the
isomorphism classes in ''K''-Vect correspond exactly to the
cardinal number
In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the cas ...
s, and that ''K''-Vect is
equivalent to the
subcategory of ''K''-Vect which has as its objects the vector spaces ''K''
''n'', where ''n'' is any cardinal number.
Generalizations
The category of
sheaves of modules over a
ringed space also has enough injectives (though not always enough projectives).
See also
*
Algebraic K-theory (the important invariant of the category of modules.)
*
Category of rings
*
Derived category
*
Module spectrum
*
Category of graded vector spaces
*
Category of representations
*
Change of rings
*
Morita equivalence
*
stable module category
*
Eilenberg–Watts theorem
References
Bibliography
*
*
*
External links
*
Vector spaces
Linear algebra
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