proper generator in a Grothendieck category
Definition 0.1
Let

be a category. A family of its objects

is said to be a
family of generators of

if for every pair of distinct morphisms

there is a morphism

for some index

such that

.
One notes that in an additive category,
is a family of generators if and only if for each nonzero morphism
in
there is a morphism
such that
.
Definition 0.2
An object

in

is called a
generator for

if

with

being a family of generators for

.
Equivalently, (viz. Mitchell)
is a generator for
if and only if the
set-valued functor
is an imbedding functor.
Definition 0.3
A
proper generator 
of a Grothendieck category

is defined as
a generator

which has the property that a monomorphism

induces an isomorphism

,
if and only if

is an isomorphism.
Theorem 0.1
Any commutative ring is the endomorphism ring of a proper generator in a suitably chosen
Grothendieck category.
Contributors to this entry (in most recent order):
As of this snapshot date, this entry was owned by bci1.