In algebra, an augmentation of an associative algebra A over a commutative ring k is a k-algebra homomorphism A → k {\displaystyle A\to 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 ] {\displaystyle A=k[G]} is the group algebra of a finite group G, then

A → k , ∑ a i x i ↦ ∑ a i {\displaystyle A\to k,\,\sum a_{i}x_{i}\mapsto \sum a_{i}}

is an augmentation.

If A is a graded algebra which is connected, i.e. A 0 = k {\displaystyle A_{0}=k}, then the homomorphism A → k {\displaystyle A\to 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 {\displaystyle k[x]\to k,\sum a_{i}x^{i}\mapsto a_{0}}

is an augmentation on the polynomial ring k [ x ] {\displaystyle k[x]}.