MathDB

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 that
a)

i\left( \mathbb{Z}_{12} \right) =4.

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

Problem Statement