Journal of the Faculty of Liberal Arts, Yamaguchi University. Natural science

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Journal of the Faculty of Liberal Arts, Yamaguchi University. Natural science Volume 29
published_at 1995

Complete reducibility in some algebraic systems

Complete reducibility in some algebraic systems
Kashiwagi Yoshimi
Kikumasa Isao
Descriptions
In some algebraic systems we find algebraic objects which are direct sums of indecomposable or simple or irreducible sub-objects. Conditions for such complete reducibility are different, but results are quite similar. Examples are the Krull-Remak-Schmidt's Theorem about the decomposition of groups with chain conditions into indecomposable subgroups and the decomposition of semi-simple Lie algebras over a field of characteristic 0 into simple ideals. We will see this phenomenon in error-correcting codes, non-associative algebras (especially Lie algebras), and modules over a commutative ring, which are completely different algebraic systems. Proofs except for error-correcting codes are traditional in this paper, but the viewpoint seems to be now.