Artin algebra

Dual and transpose

• If M is a left Λ module then the right Λ-module M^* is defined to be Hom_Λ.
• The dual D of a left Λ-module M is the right Λ-module D = Hom_R, 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.
• The transpose Tr 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.