TITLE: Question on the definition of a ring.
QUESTION [5 upvotes]: A ring $\langle R,+, \cdot\rangle $ is a set $R$ with two binary operations such that:

$\langle R,+\rangle $ is an abelian group.
Multiplication is associative.
Left and right distributive laws hold.

Can someone give me an example where the left distributive law holds but the right one doesn't?

REPLY [6 votes]: Consider the set of affine maps from $\mathbb R$ to $\mathbb R$ under addition and composition. We have right-distributivity as for affine maps $ax+b,cx+d,ex+f$ we get
$$a(cx+d)+b+e(cx+d)+f=(a+e)(cx+d)+(b+f)$$
but not left-distributivity as
$$a(cx+d+ex+f)+b\neq a(cx+d)+b+a(ex+f)+b$$
when $b\neq 0$. We can make it left-distributive but not right-distributive by simply reversing the order of composition.