Complete reducibility in some algebraic systems

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.