Category theory MOC

Category semigroup

Let be a commutative ring and be a Small category. The category ring is an -semigroup constructed from the free module . #m/def/cat This is a generalization of the Monoid ring in light of Monoids as categories. In the case is finite, this construction gives an extension ring of and is called the category ring which we denote by .

Construction

We begin with the free module taking the objects as identities convention, and linearly extend the following product for

If is finite, then this forms an -monoid with an identity given by

Properties

• Module over a category ring

Special case

• Path algebra

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