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Total algebra

Generalization of monoid ring

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Source updated January 13, 2024

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

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

and with the multiplication law given by:

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

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

These operations turn $R^{S}$ 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.

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