algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

, an Artin algebra is an

Λ over a commutative Artin ring ''R'' that is a finitely generated ''R''-module. They are named after

Emil Artin Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrians, Austrian mathematician of Armenians, Armenian descent. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number t ...

. 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 In mathematics, specifically abstract algebra, the opposite of a ring is another ring with the same elements and addition operation, but with the multiplication performed in the reverse order. More explicitly, the opposite of a ring is the ring w ...

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 Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by John ...

''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 The cokernel of a linear mapping of vector spaces is the quotient space of the codomain of by the image of . The dimension of the cokernel is called the ''corank'' of . Cokernels are dual to the kernels of category theory, hence the nam ...

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 Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...

, series=Cambridge Studies in Advanced Mathematics , volume=36 , year=1997 , isbn=978-0-521-59923-8 , mr=1314422 , zbl=0834.16001 Ring theory