The claim that there are two binary operations on rings is misleading. Rings are actually equipped with countably many $n$-ary operations, one for each noncommutative polynomial in $n$ variables over $\mathbb{Z}$. These generate the morphisms in a category with finite products, the Lawvere theory of rings $T$, which is a category with the property that finite product-preserving functors $T \to \text{Set}$ are the same thing as rings. It just happens to be the case that as a category with finite products, $T$ is generated by addition and multiplication. The Lawvere theory of commutative rings is similar except that the polynomials are commutative; incidentally, it may also be regarded as the category of affine spaces over $\mathbb{Z}$.
This gives a useful perspective from which to understand other ring-like structures. For example, :
- commutative Banach algebras are equipped with an $n$-ary operation for each holomorphic function $\mathbb{C}^n \to \mathbb{C}$, and mathbb{C}$.
- smooth algebras like the algebras $C^{\infty}(M)$ of smooth functions on a smooth manifold are equipped with an $n$-ary operation for each smooth function $\mathbb{R}^n \to \mathbb{R}$.
Here is a general procedure for determining what operations are actually available to you when working with some mathematical objects. If $C$ is a concrete category and $F : C \to \text{Set}$ the forgetful functor, then one interpretation of "$n$-ary operation" is "natural transformation $F^n \to F$." If $C$ has finite coproducts and $F$ is representable by an object $a$, then by the Yoneda lemma these are the same thing as elements of $F(a \sqcup ... \sqcup a)$. This reproduces the obvious answers for groups, rings, etc., and when $C$ is the opposite of the category of smooth manifolds and $F : M \mapsto C^{\infty}(M)$ then we get that "$n$-ary operation" means element of $C^{\infty}(\mathbb{R}^n)$ as above.
