If $R$ is an algebra without a unit, then the standard unitisation $R^\sharp$ can have maximal one-sided ideals other than $R$. Thus, it is natural to ask about the following. Let $R$ be an algebra without a unit (over a field with char 0 if it does matter).
Is there a unital algebra $A$ such that $R$ is the unique maximal right ideal in $A$?
EDIT: Assume $R$ has a faithful representation $\pi$ on a vector space $V$ (over $K$), what if we consider $A=\pi(R)+KI$?