algebra Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...

, an Artin algebra is an

Λ over a commutative

Artin ring In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are na ...

''R'' that is a finitely generated ''R''-module. They are named after

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

. 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 ...

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 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 name: ...

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 is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambr ...

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