Question about distributive law in definition of a ring

Question

In the definition of a ring $R$, one has

1. $a(b+c) = ab + ac$ and
2. $(a+b)c = ac + bc$

for all $a,b,c\in R$

My question is (just out of curiosity) if one really needs both of these. I can't think of an example of something that is not a ring that only satisfies one of the sides of the distributive law. So can one prove that if $a(b+c) = ab + ac$ for all $a,b,c$, then $(a+b)c = ac + bc$ for all $a,b,c$.

Edit: I maybe should add that all rings in my definition have a unity $1$.

Answer 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 $$p \circ (q_1 + q_2) \color{red} \neq p \circ q_1 + p \circ q_2.$$