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Groupprops β

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

Related facts

Proof

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