Ring of functions

If R is a ring, and S any nonempty set, then the set of all functions from S to R, also denoted by R^S, has again the structure of a ring for the operations defined by

$f+g:={\begin{cases}S\to R\\x\mapsto f(x)+g(x)\end{cases}}$

and

$f\cdot g:={\begin{cases}S\to R\\x\mapsto f(x)\cdot g(x)\end{cases}}$.

Examples

In particular, for $S=\mathbb {N}$, this yields the usual ring of sequences with "pointwise" multiplication and addition, i.e., $(u_{i})\cdot (v_{i})=(u_{i}\,v_{i})$ and $(u_{i})+(v_{i})=(u_{i}+v_{i})$.

For a finite set S with n elements, e.g., S = { 0,..., n-1 }, the ring R^S can be identified with Rⁿ ("= R × ... × R", where quotes indicate that this notation is usually ill-defined...), in the sense that $x_{i}=x(i)$ is seen as the i-th component of the "n-component vector" x (which actually is a function). (When the natural numbers are defined in an axiomatic set-theoretical way, then n is often defined as { 0, 1, ..., n-1 }, namely, one defines 0 = {} and n+1 = { n } union n. With this definition, R^S is actually identical to Rⁿ.)

Such "product rings" always have divisors of zero, as soon as #S = card(S) > 1. (Any two nonzero elements which have their nonzero components at different indices, will have a vanishing product.)

Other constructions

There are several other constructions of more complex rings, based on a given ring R, which do not have this "problem", in particular

• the ring of polynomials, finitely supported R-valued sequences with the convolution product,
• Cayley-Dickson algebras, such as the complex numbers, where R × R is equipped with a multiplication $(x_{1},x_{2})\cdot (y_{1},y_{2}):=(x_{1}\,y_{1}-y_{2}^{*}\,x_{2},y_{2}\,x_{1}+x_{2}\,y_{1}^{*})$. (The stars refer to a possibly existing involution on R and can be ignored if R is "real".)