I have a non-empty set $R$ with a binary operation $*$. If the pair has an identity element $e\in R$ and $$(a*b)*(c*d)=(a*c)*(b*d)$$
holds for all $a,b,c,d \in R$, how do I then prove that this is associative?
I understand the concept that $x*(y * z) = (x * y)*z$, but I am unsure how to apply it here.
For $b=e$ it follows follows that $a\cdot b=a$ and $b\cdot d=d$ so that $$ (a\cdot b)\cdot(c\cdot d)=a\cdot (c\cdot d), \; (a\cdot c)\cdot(b\cdot d)=(a\cdot c)\cdot d. $$ Hence $a\cdot (c\cdot d)=(a\cdot c)\cdot d$ for all $a,c,d$, which means associativity.