Artin algebra
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(Redirected from Artinian algebra)
In algebra, an Artin algebra is an algebra
Every Artin algebra is an Artin ring.
Dual and transpose
[edit]There are several different dualities taking finitely generated modules over
- If M is a left
Λ -module then the rightΛ -module M* is defined to be HomΛ (M,Λ ). - The dual D(M) of a left
Λ -module M is the rightΛ -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 overΛ does not depend on the choice of R (up to isomorphism). - The transpose Tr(M) of a left
Λ -module M is a rightΛ -module defined to be the cokernel of the map Q* → P*, where P → Q → M → 0 is a minimal projective presentation of M.
References
[edit]- Auslander, Maurice; Reiten, Idun; Smalø, Sverre O. (1997) [1995], Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, ISBN 978-0-521-59923-8, MR 1314422, Zbl 0834.16001