Do the rng axioms enforce any hidden restrictions (except the existence of an absorbing element) on the multiplication operator? Definition. An operation $*$ on a set $R$ has an absorbing element iff there exists an element $0 \in R$ such that for all $x \in R$ it holds that $0*x=x*0=0$.
Thus, if $(R,+,*)$ is a rng, then $*$ has an absorbing element, namely the additive identity.
Question 1. Does assuming that $(R,+,*)$ is an rng imply any other statements about $*$, other than those implied by associativity together with the existence of an absorbing element?
We can also recast this proof-theoretic question into a model-theoretic question.
Question 2. Is it true that for all sets $R$ and all associative binary operations $*$ on $R$ such that $*$ has an absorbing element, there exists a binary operation $+$ on $R$ such that $(R,+,*)$ is an rng?
N.B. Let us not assume that the rng is commutative.

Counter-example: $R = \{ 0,1,2,3,4 \}$ and $x*y = \text{lcm}(x,y) \,$ (mod 5). Then $0 * x = x * 0 = 0$ for all $x \in R.$ You can't put an abelian group structure on $R$ because the only candidate is the cyclic group of order 5, and it's easy to check that doesn't obey the distributive laws. 
