Let $M$ be the set of all $3\times 2$ matrices over a field $F_2$ with usual addition. Let $P$ be the $2\times 3$ matrix consisting of $1$'s. Define a multiplication: $A*B= APB$. Then we get a ring (this is so-called Munn algebra) with required properties (a minimal left ideal is generated by a column).

In such a way you can obtain a ring with $m$ minimal left ideals and $n$ minimal right ideals.

Let $M$ be the set of all $3\times 2$ matrices over a field $F_2$ with usual addition. Let $P$ be the $2\times 3$ matrix consisting of $1$'s. Define a multiplication: $A*B= APB$. Then we get a ring (this is so-called Munn algebra) with required properties (a minimal left ideal is generated by a column).

In such a way you can obtain a ring with $m$ minimal left ideals and $n$ minimal right ideals.

Addendum (Thanks to Manny Reyes) $M$ doesn't contain an identity element, but we can join an identity $e$ to $M$, i.e. consider the ring $M_1$ of pairs $(a,me)$ ($a\in M, m\in F_2$) with multiplication $(a,me)*(b,ne)=(a*b+na+mb,mne)$. Then minimal ideals of $M_1$ are the same of $M$.