# Artin algebra

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*.

## References

- Auslander, Maurice; Reiten, Idun; Smalø, Sverre O. (1997) [1995],
*Representation theory of Artin algebras*, Cambridge Studies in Advanced Mathematics,**36**, Cambridge University Press, ISBN 978-0-521-59923-8, MR 1314422, Zbl 0834.16001

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