Augmentation (algebra)

In algebra, an augmentation of an associative algebra "A" over a commutative ring "k" is a "k"-algebra homomorphism formula_1, 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 formula_2 is the group algebra of a finite group "G", then
is an augmentation. 
If "A" is a graded algebra which is connected, i.e. formula_4, then the homomorphism formula_5 which maps an element to its homogeneous component of degree 0 is an augmentation. For example,
is an augmentation on the polynomial ring formula_7.