TITLE: Is there a finite ring whose rank is smaller than the rank of its group and its monoid?
QUESTION [9 upvotes]: 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?

REPLY [4 votes]: Let me generalize Pierre-Guy Plamondon's example.  Let $p$ be a prime and let $R=\mathbb{F}_p^n$.  Then the additive rank of $R$ is $n$.  If $x\in R$ has exactly one coordinate which is $0$ and $x=yz$, then at least one of $y$ and $z$ must have that same coordinate as its only coordinate which is $0$.  Thus the multiplicative rank is at least $n$, since any generating set must contain an element which has exactly one coordinate which is zero, for each possible coordinate.  (For $p=2$ the rank is exactly $n$; for $p>2$ it will be $2n$, since the group of units has rank $n$, and any set of multiplicative generators must also contain a set of generators for the units.)  But I claim the rank of $R$ as a ring is only $\lceil\log_p n\rceil$.   For each $i<\lceil\log_p n\rceil$ let $x_i\in R$ be the element whose $k$th coordinate is the $i$th base $p$ digit of $k$.  Then the set $\{x_i:0\leq i<\lceil\log_p n\rceil\}$ separates coordinates, and it follows that it generates $R$ as a ring (in general, if $K$ is a field, then (unital) $K$-subalgebras of $K^n$ are determined by the coordinates they separate).  Conversely, it is easy to see that no subset of $R$ of cardinality less than $\lceil\log_p n\rceil$ can separate coordinates, and thus no smaller set can generate the ring.
In particular, taking $p\geq n$, this gives rings which are generated by a single element as a ring but require arbitrarily large numbers of elements to generate either additively or multiplicatively.