Augmentation (algebra)
In algebra, an augmentation of an associative algebra A over a commutative ring k is a k-algebra homomorphism, 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 is the group algebra of a finite group G, then
is an augmentation.
If A is a graded algebra which is connected, i.e., then the homomorphism which maps an element to its homogeneous component of degree 0 is an augmentation. For example,
is an augmentation on the polynomial ring.
