Question about distributive law in definition of a ring

Solution 1:

Here is an example that fails precisely in left distributivity.

Consider $\mathbb{R}[X]$ - the polynomials with coefficients from $\mathbb{R}$ with the usual operation of pointwise addition (in fact, the ring of scalars is irrelevant here).

The tricky part is how we define multiplication: let $p \cdot q$ be the composition $p \circ q$. This multiplication is associative, and even has an identity, which is the identity polynomial $p(x)=x$.

Now, trivially $$(p_1 + p_2) \circ q = p_1 \circ q + p_2 \circ q,$$ but in general $$p \circ (q_1 + q_2) \color{red} \neq p \circ q_1 + p \circ q_2.$$