Total algebra

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 $$(t,u)\in S\times S$$ 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:
 * $$(\alpha\cdot\beta)(s) = \sum_{tu=s}\alpha(t)\beta(u).$$

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.