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

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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.

Facts used

  1. Associativity is a groupy relation on the additive group of a non-associative ring
  2. Fraction of tuples satisfying groupy relation in subgroup is at least as much as in whole group

Related facts

Proof

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