abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...

, the total algebra of a

monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. Monoids ...

is a generalization of the

monoid ring In abstract algebra, a monoid ring is a ring constructed from a ring and a monoid, just as a group ring is constructed from a ring and a group. Definition Let ''R'' be a ring and let ''G'' be a monoid. The monoid ring or monoid algebra of ''G'' ...

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_\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 In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial s ...

, where the monoid ''S'' is the

natural numbers In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...

. The product is then the

Cauchy product In mathematics, more specifically in mathematical analysis, the Cauchy product is the discrete convolution of two infinite series. It is named after the French mathematician Augustin-Louis Cauchy. Definitions The Cauchy product may apply to infini ...

References

* {{citation, author= Nicolas Bourbaki, title=Algebra, publisher=Springer, year=1989: §III.2 Abstract algebra