Characteristic property of the direct sum module

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Statement


TODO: Caption


Let (R,+,,0) be a ring (with or without unity) and let (Mα)αI be an arbitrary indexed family of R-modules and αIMα their direct sum (external or internal). Let M be another R-module. Then[1]:
  • For any family of module homomorphisms, (φ:MαM)αI
    • There exists a unique module homomorphism, φ:αIMαM, such that
      • αI[φiα=φα]

TODO: Mention commutative diagram and such



Proof

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Should be routine enough, see page 327 in Abstract Algebra - Grillet if stuck

Notes

References

  1. Jump up Abstract Algebra - Pierre Antoine Grillet