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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 ∈ S {\displaystyle s\in S} , there exist only finitely many ordered pairs ∈ S × S {\displaystyle \in S\times S} for which t u = s {\displaystyle tu=s}. Let R be a ring. Then the total algebra of S over R is the set R S {\displaystyle R^{S}} of all functions α : S → R {\displaystyle \alpha :S\to R} 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 R S {\displaystyle R^{S}} into a ring. There is an embedding of R into R S {\displaystyle R^{S}} , given by the constant functions, which turns R S {\displaystyle R^{S}} into an R-algebra.