RO-NMO 2003 Problem 124

$i(L)$ denotes the number of multiplicative binary operations over the set of elements of the finite additive group $L$ such that the set of elements of $L,$ along with these additive and multiplicative operations, form a ring. Prove thata) $i\left( \mathbb{Z}_{12} \right) =4.$ b) $i(A\times B)\ge i(A)i(B) ,$ for any two finite commutative groups $B$ and $A.$ c) there exist two sequences $\left( G_k \right)_{k\ge 1} ,\left( H_k \right)_{k\ge 1}$ of finite commutative groups such that $$\lim_{k\to\infty }\frac{\# G_k }{i\left( G_k \right)} =0$$ and $$\lim_{k\to\infty }\frac{\# H_k }{i\left( H_k \right)} =\infty.$$ Barbu Berceanu