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, there exist only finitely many ordered pairs for which.
Let R be a ring. Then the total algebra of S over R is the set of all functions with the addition law given by the operation:
and with the multiplication law given by:
The sum on the right-hand side has finite support, and so is well-defined in R.
These operations turn into a ring. There is an embedding of R into, given by the constant functions, which turns 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.
