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Let A be a finite-dimensional algebra. Prove that A is a semisimple algebra if and only if every left ideal I of A admits a complement (so there exists a left ideal J such that A = I J as vector spaces). You I may assume that if a module M can be written as a sum of simple modules M = S₁+ S₂++ Sn, then we can find a subset R C {1, 2, ………, n} such that DER ST is a direct sum of simples. You may also assume that every non-zero finite-dimensional module has a simple submodule. M =

Let A be a finite-dimensional algebra. Prove that A is a semisimple algebra if and only if every left ideal I of A admits a complement (so there exists a left ideal J such that A = I J as vector spaces). You I may assume that if a module M can be written as a sum of simple modules M = S₁+ S₂++ Sn, then we can find a subset R C {1, 2, ………, n} such that DER ST is a direct sum of simples. You may also assume that every non-zero finite-dimensional module has a simple submodule. M =

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Vector Spaces
Section4.2: Vector Spaces
Problem 44E: Prove that in a given vector space V, the additive inverse of a vector is unique.
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Let A be a finite-dimensional algebra. Prove that A is a semisimple algebra if and only if every left ideal I of A admits a complement (so there exists a left ideal J such that A = I J as vector spaces). You I may assume that if a module M can be written as a sum of simple modules M = S₁+ S₂++ Sn, then we can find a subset R C {1, 2, ………, n} such that DER ST is a direct sum of simples. You may also assume that every non-zero finite-dimensional module has a simple submodule. M =
Transcribed Image Text:Let A be a finite-dimensional algebra. Prove that A is a semisimple algebra if and only if every left ideal I of A admits a complement (so there exists a left ideal J such that A = I J as vector spaces). You I may assume that if a module M can be written as a sum of simple modules M = S₁+ S₂++ Sn, then we can find a subset R C {1, 2, ………, n} such that DER ST is a direct sum of simples. You may also assume that every non-zero finite-dimensional module has a simple submodule. M =
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