Category theory MOC Category semigroup Let π be a commutative ring and π’ be a Small category. The category ring π π’ is an R-semigroup constructed from the free module π (π’). cat This is a generalization of the Monoid ring in light of Monoids as categories. In the case Obβ‘(π’) is finite, this construction gives an extension ring of π and is called the category ring which we denote by π [π’]. Construction We begin with the free module π (π’) taking the objects as identities convention, and linearly extend the following product for π,π βπ’ πβ π={0codβ‘πβ domβ‘ππβπcodβ‘π=domβ‘π. If Obβ‘(π’) is finite, then this forms an R-monoid with an identity given by 1=βπ₯βObβ‘(π’)1π₯. Properties Module over a category ring Special case Path algebra develop | en | SemBr