Let $R$ be a ring and $A$ be an $R$-algebra and $B\subset A$.
What condition should $A,B,R$ have to get the concept of quotient $R$-algebra? So that we can define $(a+B)+(b+B)=(a+b)+B$ and $(a+B)(b+B)=(ab)+B$ and $r(a+B)=(ra)+B$ and $A/B$ is an $R$-algebra under this operation?
I think I should assume that $A$ is an associative algebra and $B$ is an ideal of $A$, viewing $A$ as a rng and $B$ is an $R$-submodule of $A$.
You should ensure that $B$ is a two-sided ideal, so that you can also get the law $(a+B)r = (ar) + B$. It should not be an $R$-submodule, but an $R$-subbimodule.