I know that it is impossible to axiomatize the multiplicative structures of rings, called $R$-semigroups. Is anything known about the first-order axiomatizability of the class of abelian groups which are additive groups of some ring? I don't want to restrict the meaning "ring" here. I would like to know whether this question is answered for rings with any subset of the set of adjectives {"associative", "unitary", "commutative"} attached.

**EDIT** I forgot to mention that I do want to exlude *some* rings, that is rings with zero multiplication.