Turning a subalgebra into an ideal by changing the algebra it is contained in

Question

Let $B$ be an (not necessarily unital) algebra and $A \subseteq B$ a subalgebra. Is there some sort of quotient $C$ of $B$ (or any other interesting construction not necessarily found by taking quotients, but with perhaps some nice universal property, while being as similar to $B$ as possible) such $A$ is still contained in $C$, but $A$ is, further, an ideal of $C$?