# What is “augmented algebra”?

Really sorry for this question, but googling for some time did not help me. I was trying to understand the meaning of the following phrase:

Let B be an augmented algebra over a semi-simple algebra T.

But I am stuck already with "augmented algebra"... -- can not find a definition on the web.

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The notion of an augmented algebra is defined, for example, in Mac Lane's book Homology. There an augmented algebra $A$ over a field $k$ is a $k$-algebra that is also equipped with an algebra homomorphism $A \rightarrow k$ (the augmentation map). So in your context, I would assume that $B$ is a $T$-algebra that is also equipped with an algebra homomorphism $B \rightarrow T$. – Christopher Drupieski Feb 12 '12 at 0:16
It probably means that there is a map $\iota : T \to B$ of rings ($B$ is a $T$-algebra) which admits a splitting $\epsilon : B \to T$ (i.e. $\epsilon \circ \iota = 1_T$) – Konstantin Ardakov Feb 12 '12 at 0:18

An augmented ring is simply a triple $(A,M,\epsilon)$ with $A$ ring, $M$ a left $A$-module and $\epsilon:A\to M$ a surjection of $A$ modules. One says then that $A$ is augmented over $M$. You can find this defined in Cartan-Eilenberg, for example.
Often, $M$ is itself a ring and the map $\epsilon$ is also a ring morphism. This is probably the situation you have.
C-E also define supplemented algebras: a supplemented $k$-algebra is a $k$-algebra augmented over $k$ with $k$-linear augmentation. This is the concept that appears in the comments above. The name comes from the fact that in a supplemented algebra $A$ the base ring $k$ is a summand with a supplement, namely the kernel of the augmentation. – Mariano Suárez-Alvarez Feb 12 '12 at 0:23