In algebra, an augmentation of an associative algebra A over a commutative ring k is a k-algebra homomorphism
A
→
k
, typically denoted by ε. An algebra together with an augmentation is called an augmented algebra. The kernel of the augmentation is a two-sided ideal called the augmentation ideal of A.
For example, if
A
=
k
[
G
]
is the group algebra of a group G, then
A
→
k
,
∑
a
i
x
i
↦
∑
a
i
is an augmentation.
If A is a graded algebra which is connected, i.e.
A
0
=
k
, then the homomorphism
A
→
k
which maps an element to its homogeneous component of degree 0 is an augmentation. For example,
k
[
x
]
→
k
,
∑
a
i
x
i
↦
a
0
is an augmentation.
