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, #m/def/ring as formalized by the Universal property. Thus it is an -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

https://q.uiver.app/#q=WzAsNCxbMCwwLCJSIl0sWzIsMCwiUltNXSJdLFs0LDAsIk0iXSxbMiwyLCJUIl0sWzAsMSwiXFxpb3RhIl0sWzIsMSwiXFxtdSIsMl0sWzAsMywiSSIsMl0sWzIsMywibSJdLFsxLDMsIlxcZXhpc3RzICFmIiwxLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV1d

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

#state/tidy | #lang/en | #SemBr

Footnotes

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