Left-associative elements of non-associative ring form associative subring

Statement
Suppose $$S$$ is a non-associative ring (i.e., a not necessarily associative ring). Suppose $$L$$ is the subset of $$S$$ comprising those elements $$x \in S$$ such that:

$$\! x * (y * z) = (x * y) * z \ \forall y,z \in S$$.

Then, $$L$$ is closed under the ring operations of $$S$$ and is an associative ring with the induced operations. Further:


 * If $$S$$ contains a multiplicative identity, $$L$$ contains that identity.
 * If an element of $$L$$ has a two-sided inverse in $$S$$, that two-sided inverse in fact lies inside $$L$$.

For middle and right associativity

 * Middle-associative elements of non-associative ring form associative subring
 * Right-associative elements of non-associative ring form associative subring

Analogous statements for magmas and loops

 * Left-associative elements of magma form submagma
 * Middle-associative elements of magma form submagma
 * Right-associative elements of magma form submagma
 * Left-associative elements of algebra loop form subgroup
 * Middle-associative elements of algebra loop form subgroup
 * Right-associative elements of algebra loop form subgroup