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 $\mathbb{Z}$.
• 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)$as above. 2 added 721 characters in body; added 2 characters in body The claim that there are two binary operations on rings is misleading. Rings are actually equipped with countably many$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}$, and smooth algebras like the algebras$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)$as above. 1 The claim that there are two binary operations on rings is misleading. Rings are actually equipped with countably many$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}$, and smooth algebras like the algebras$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}\$.