Total algebra explained

In abstract algebra, the total algebra of a monoid is a generalization of the monoid ring that allows for infinite sums of elements of a ring. Suppose that S is a monoid with the property that, for all

s\in S , there exist only finitely many ordered pairs (t,u)\in S x S for which tu = s . Let R be a ring. Then the total algebra of S over R is the set R ^{S} of all functions \alpha:S\to R with the addition law given by the (pointwise) operation:(\alpha + \beta)(s) = \alpha(s) + \beta(s) and with the multiplication law given by:(\alpha ⋅ \beta)(s) = \sum _{tu=s} \alpha(t)\beta(u). The sum on the right-hand side has finite support, and so is well-defined in R.

These operations turn

R ^{S} into a ring. There is an embedding of R into R ^{S} , given by the constant functions, which turns R ^{S} into an R-algebra.

An example is the ring of formal power series, where the monoid S is the natural numbers. The product is then the Cauchy product.

References

• : §III.2