The operation of ring multiplication is axiomatizable, if we allow ourselves an additional auxillary addition symbol. Just write down the ring axioms in the signature $\{*,+\}$. But could multiplication be axiomatizable by itself, without an auxillary aid?
That is to say, consider the class of structures $(R, *)$ such that there exists an operation $+$ that makes $(R, +, *)$ into a ring. Is that class axiomatizable in the signature $\{*\}$?
The more important question is, is there a standard term in the logic literature for a class of relations or operations which may not necessarily be axiomatizable by themselves, but can be so with the aid of auxillary relations and/or operations? I am sorry if my questions sound a little confusing at first.
The first Question: "That is to say, consider the class of structures $(R, *)$ such that there exists an operation $+$ that makes $(R, +, *)$ into a ring. Is that class axiomatizable in the signature $\{*\}$?"
The Answer: "No." This is proved in
Kogalovskiĭ, S. R. The multiplicative semigroups of rings. (Russian) Dokl. Akad. Nauk SSSR 140 1961 1005–1007.