In this chapter we turn our attention to artin algebras and their finitely generated modules, the main subject of this book. One important feature of the theory of artin algebras as opposed to left artin rings is that endomorphism rings of finitely generated modules are again artin algebras. In principle, this enables one to convert problems involving only a finite number of modules over one artin algebra to problems about finitely generated projective modules over some other artin algebra. This procedure, which we call projectivization, is illustrated by our proofs of the Krull–Schmidt theorem and other results. Another important property of artin algebras is that there is a duality between finitely generated left and finitely generated right modules. It is convenient to start the chapter with a section on categories over a commutative artin ring R, and study equivalences of such categories.

Artin algebras and categories

Generalizing the category mod Λ for an artin R-algebra Λ we introduce the notion of R-categories, and study equivalences between such categories.

Let R be a commutative artin ring. We recall that an R-algebra Λ is a ring together with a ring morphism ϕ: R → Λ whose image is in the center of Λ. For an R-algebra ϕ: R → Λ we usually write rλ for ϕ(r)λ where r is in R and λ is in Λ. If ϕ1:R → Λ1 and ϕ2:R → Λ2 make Λ1 and Λ2R-algebras, then Λ1 is an R-subalgebra of Λ2 if it is a subring of Λ2 via i: Λ1 → Λ2 and iϕ1 = ϕ2.