total algebra

{{Short description|Generalization of monoid ring}}

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 $s\backslash in\; S$, there exist only finitely many ordered pairs $(t,u)\backslash in\; S\backslash times\; S$ 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 $\backslash alpha:S\backslash to\; R$ with the addition law given by the (pointwise) operation:

:$(\backslash alpha+\backslash beta)(s)=\backslash alpha(s)+\backslash beta(s)$

and with the multiplication law given by:

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

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.