The class of algebras that can be decomposed into the sum of a two-sided ideal and the algebra generated by the unit

Question

Let $A$ be an associative algebra, say, over $\mathbb C$, and let $1_A$ be its unit. Can anybody enlighten me what it is called when there exists a two-sided ideal $I$ in $A$ such that $A$ is decomposed, as a vector space over $\mathbb C$, into the direct sum $$ A=I\oplus {\mathbb C}\cdot 1_A\ ? $$ I think, this class of algebras must have a name, I wonder what is known about it. Thank you.

Answer 1

These are precisely the algebras admitting a homomorphism to the base. Such algebras (together with the choice of homomorphism, which is sometimes called an augmentation) are called augmented algebras. A natural source of examples is given by bialgebras.