Ring theory MOC

Monoid ring

Let be a ring and be a monoid¹. The monoid ring is the extension ring of by adjoining in the most general way maintaining the monoid product as ring multiplication, ring as formalized by the Universal property. Thus it is an R-monoid constructed from the free module .

Universal property

Let be a ring and be a monoid. The associated monoid ring is a triple consisting of a ring , a ring homomorphism , and a monoid homomorphism ; such that given any ring , ring homomorphism , and monoid homomorphism there exists a unique ring homomorphism such that the following commutes in

This admits a unique extension to a bifunctor such that

become natural transformations.

Construction as maps

As with the free module, may be constructed as the set of maps of finite support , where we identify with invoking an Iverson bracket, and elements of with constant functions. For , the product is given by

Proof of universal property

Clearly is an abelian group under pointwise addition. The convolution operation is associative, since

and a multiplicative identity is given by . Clearly is a Ring monomorphism, and is a monoid monomorphism since

Now suppose is another such triple. For the diagram to commute, we require that for all and that for all . For to be a ring homomorphism, it follows

and thus for

which is unique, as required.

See also

• Group ring

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Footnotes

1. Or a semigroup, where one simply uses its completion to a monoid. ↩