Augmentation (algebra)

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In algebra, an augmentation of an associative algebra A over a field k is an algebra homomorphism A \to k, typically denoted by ε. An algebra together with an augmentation is called an augmented algebra.

For example, if A =k[G] is the group algebra of a group G, then

A \to k, \, \sum a_i x_i \mapsto \sum a_i

is an augmentation. The kernel is called the augmentation ideal of A.