Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Linear applications simple doubt

  1. Oct 22, 2013 #1
    Is an injective endomorfism automatically bijective?
  2. jcsd
  3. Oct 22, 2013 #2
  4. Oct 23, 2013 #3
    Thanks for confirming that intuition and for the pointer to Fredholm alternative, never heard of it.
  5. Oct 24, 2013 #4


    User Avatar
    Science Advisor
    Gold Member

    Doesn't this follow from rank-nullity theorem?
  6. Oct 25, 2013 #5


    User Avatar
    Science Advisor
    Homework Helper

    for finitely generated modules over other rings this can fail, e.g. if Z is the integers, the injective endomorphism Z-->Z taking n to 3n is injective but not surjective. Interestingly however, a surjective endomorphism of a finitely generated module is always injective as well.
  7. Oct 25, 2013 #6
    Yes. But the Fredholm alternative also hold in some infinite-dimensional cases. There, you can't use rank-nullity.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook