Let $k$ be a field, and $A$ an algebra. I keep seeing reference to $k \cdot 1_A$; what does this mean?
For example, if $A$ is an augmented algebra, via $\epsilon : A \to k$, then $A$ is canonically isomorphic to $k\cdot 1_A \oplus \ker \epsilon$.
I would like to prove the above, but I haven't seen the notation $k \cdot 1_A$ defined.
I think you have seen the notation before, at least in other contexts. Like $Hg$ when talking about the (right) cosets of a subgroup $H$ in some group $G\ni g$. Or when we write $Ra$ for the (left) ideal generated by an element $a$ in a ring $R$.
The notation $k\cdot 1_A$ means $$ \{x\cdot 1_A\mid x\in k\} $$ Intuitively, it's the image of the inclusion map $k\to A$.