Artin algebra explained

In algebra, an Artin algebra is an algebra Λ over a commutative Artin ring R that is a finitely generated R-module. They are named after Emil Artin.

Every Artin algebra is an Artin ring.

Dual and transpose

There are several different dualities taking finitely generated modules over Λ to modules over the opposite algebra Λ^op.

• If M is a left Λ-module then the right Λ-module M^* is defined to be Hom_Λ(M,Λ).
• The dual D(M) of a left Λ-module M is the right Λ-module D(M) = Hom_R(M,J), where J is the dualizing module of R, equal to the sum of the injective envelopes of the non-isomorphic simple R-modules or equivalently the injective envelope of R/rad R. The dual of a left module over Λ does not depend on the choice of R (up to isomorphism).
• The transpose Tr(M) of a left Λ-module M is a right Λ-module defined to be the cokernel of the map Q^* → P^*, where P → Q → M → 0 is a minimal projective presentation of M.