# Associating fraction in subring of finite non-associative ring is at least as much as in whole ring

## Statement

Suppose $R$ is a finite Non-associative ring (?) (i.e., $R$ is a not necessarily associative ring whose underlying set is finite). Suppose $S$ is a subring of $R$. Then, the associating fraction of $S$ is at least as much as that of $R$.

In symbols, if $AT(R) := \{ (x,y,z) \in R^3 \mid (x * y) * z = x * (y * z) \}$ and $AT(S) = S^3 \cap CP(R)$, then:

$\frac{|AT(S)|}{|S|^3} \ge \frac{|AT(R)|}{|R|^3}$

In fact, the result also holds if $S$ is simply an additive subgroup of $R$ and not a subring.

## Proof

The proof follows from facts (1) and (2).