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

*{{Citation , last1=Auslander , first1=Maurice , last2=Reiten , first2=Idun , last3=Smalø , first3=Sverre O. , title=Representation theory of Artin algebras , origyear=1995 , url=https://books.google.com/books?isbn=0521599237 , publisher= Cambridge University Press , series=Cambridge Studies in Advanced Mathematics , volume=36 , year=1997 , isbn=978-0-521-59923-8 , mr=1314422 , zbl=0834.16001

Ring theory