Let $F$ be a field and $A$ an $F$-algebra. (And assume that $A$ is finite dimension over $F$ if necessary.) A textbook says that $A$ is simple if it has no proper two-sided ideals.
To understand this definition well, I'm looking for an example of $F$-algebra which has no proper two-sided ideals but has one-sided ideal(s). But I cannot find out such one. Is there such example?