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 [math]\displaystyle{ s\in S }[/math], there exist only finitely many ordered pairs [math]\displaystyle{ (t,u)\in S\times S }[/math] for which [math]\displaystyle{ tu=s }[/math]. Let R be a ring. Then the total algebra of S over R is the set [math]\displaystyle{ R^S }[/math] of all functions [math]\displaystyle{ \alpha:S\to R }[/math] with the addition law given by the (pointwise) operation:
- [math]\displaystyle{ (\alpha+\beta)(s)=\alpha(s)+\beta(s) }[/math]
and with the multiplication law given by:
- [math]\displaystyle{ (\alpha\cdot\beta)(s) = \sum_{tu=s}\alpha(t)\beta(u). }[/math]
The sum on the right-hand side has finite support, and so is well-defined in R.
These operations turn [math]\displaystyle{ R^S }[/math] into a ring. There is an embedding of R into [math]\displaystyle{ R^S }[/math], given by the constant functions, which turns [math]\displaystyle{ R^S }[/math] 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
- Nicolas Bourbaki (1989), Algebra, Springer: §III.2
 |  |
