I was reading a post on mathoverflow about why we sometimes need rngs (or pseudo-rings). A contributor commented on this post an argument (here) that rings need an identity to achieve total associativity (that every possible association of a product is the same). Here is the author's argument:
But I found this argument confusing: the author is apparently assuming that $a$ is the product of something and $a$ ($()\cdot a$, which thus gives that $()$ should be the identity). However, $a$ is not required to be a product; it can be a single element, and still, the equation $a=a$ suffices by the definition of the equivalence relation $=$.
Can somebody explain this argument (or theorem) for me?