Artin algebra
Encyclopedia
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
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Every Artin algebra is an Artin ring.
Emil Artin
Emil Artin was an Austrian-American mathematician of Armenian descent.-Parents:Emil Artin was born in Vienna to parents Emma Maria, née Laura , a soubrette on the operetta stages of Austria and Germany, and Emil Hadochadus Maria Artin, Austrian-born of Armenian descent...
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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 Λop.- If M is a left Λ module then the right Λ-module M* is defined to be HomΛ(M,Λ).
- The dual D(M) of a left Λ-module M is the right Λ-module D(M) = HomR(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 cokernelCokernelIn mathematics, the cokernel of a linear mapping of vector spaces f : X → Y is the quotient space Y/im of the codomain of f by the image of f....
of the map Q* → P*, where P → Q → M → 0 is a minimal projective presentation of M.