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 ${\displaystyle s\in S}$, there exist only finitely many ordered pairs ${\displaystyle (t,u)\in S\times S}$ for which ${\displaystyle tu=s}$. Let R be a ring. Then the total algebra of S over R is the set ${\displaystyle R^{S}}$ of all functions ${\displaystyle \alpha :S\to R}$ with the addition law given by the (pointwise) operation:

${\displaystyle (\alpha +\beta )(s)=\alpha (s)+\beta (s)}$

and with the multiplication law given by:

${\displaystyle (\alpha \cdot \beta )(s)=\sum _{tu=s}\alpha (t)\beta (u).}$

The sum on the right-hand side has finite support, and so is well-defined in R.

These operations turn ${\displaystyle R^{S}}$ into a ring. There is an embedding of R into ${\displaystyle R^{S}}$, given by the constant functions, which turns ${\displaystyle R^{S}}$ 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.