symmetric monoidal (∞,1)-category of spectra
For $R$ a ring, an associative algebra over $R$ is a ring $A$ equipped with a ring inclusion $R \hookrightarrow A$.
If the $R$-algebra $A$ is equipped with an $R$-algebra homomorphism the other way round,
then it is called an augmented algebra.
In Cartan-Eilenberg this is called a supplemented algebra.
The kernel of $\epsilon$ is called the corresponding augmentation ideal in $A$.
An augmentation of a bare ring itself, being an associative algebra over the ring of integers $\mathbb{Z}$, is a ring homomorphism to the integers
Every group algebra $R[G]$ is canonically augmented, the augmentation map being the operation that forms the sum of coefficients of the canonical basis elements.
If $X$ is a variety over an algebraically closed field $k$ and $x\in X(k)$ is a closed point, then the local ring $\mathcal{O}_{X,x}$ naturally has the structure of an augmented $k$-algebra. The augmentation map $\mathcal{O}_{X,x}\rightarrow k$ is the evaluation map, and the augmentation ideal is the maximal ideal of $\mathcal{O}_{X,x}$.