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Ring of functions

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

and

.

Examples

In particular, for , this yields the usual ring of sequences with "pointwise" multiplication and addition, i.e., and .

For a finite set S with n elements, e.g., S = { 0,..., n-1 }, the ring RS can be identified with Rn ("= R × ... × R", where quotes indicate that this notation is usually ill-defined...), in the sense that 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, RS 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 . (The stars refer to a possibly existing involution on R and can be ignored if R is "real".)