# Commuting 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 commuting fraction of $S$ is at least as much as that of $R$.

In symbols, if $CP(R) := \{ (x,y) \in R^2 \mid x * y = y * x \}$ and $CP(S) = S^2 \cap CP(R)$, then:

$\frac{|CP(S)|}{|S|^2} \ge \frac{|CP(R)|}{|R|^2}$

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

## Facts used

1. Fraction of tuples satisfying groupy relation in subgroup is at least as much as in whole group

## Proof

The proof follows from fact (1), and the observation that the relation of commuting is groupy in both variables: in the sense that if we fix $x$, the set of $y$ that commute with $x$ form a subgroup of the additive group of $R$. (Note that since the relation is symmetric, groupiness in one variable is equivalent to groupiness in the other).