Consider a finite ring $(R, +, \times)$ comprising a finite additive abelian group $(R, +)$, a finite multiplicative monoid $(R, \times)$, and a distributivity rule relating the two. Let the rank of the ring be the minimum size of a set that generates the ring (that is, a set $S$ of ring elements such that the closure of $S$ under both addition and multiplication is $R$). The rank of this ring is bounded above by the smaller of the rank of the additive group and the rank of the multiplicative monoid, since a generating set for either the group or the monoid is a generating set for the ring.
Is there a finite ring whose rank is strictly less than both the rank of its group and the rank of its monoid? In other words, is there a ring in which the combination of both addition and multiplication allows us to generate more elements than using either operation individually?