Let $S$ be a finite commutative semigroup with identity. Under what conditions (on the semigroup $S$) it is possible to find a ring $R$ such that the multiplicative structure of $R  \{0\}$ is isomorphic to $S$?

$S$ must be a cyclic group of order $p^n1$ for some prime $p$ and natural $n$. Indeed, since $R\setminus \{0\}$ is a semigroup under multiplication, $R$ does not have zero divisors. Hence $R$ is a division ring. Since $S$ is finite, $R$ is a finite division ring, hence, by Wedderburn, a finite field. Therefore $S$ must be the multiplicative group of a finite field, hence a cyclic group of order $p^n1$. Note that you do not need to assume that $S$ is commutative. 

