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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 {\displaystyle A=k} is the group algebra of a finite group G, then
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,