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 Λop.
- If M is a left Λ module then the right Λ-module M* is defined to be HomΛ(M,Λ).
- The dual D(M) of a left Λ-module M is the right Λ-module D(M) = HomR(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.