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Basic ring theory

  • 04-04-2007 5:18pm
    #1
    Fremen
    Registered Users, Registered Users 2 Posts: 2,481 ✭✭✭


    Hi,
    there's a question on one of my past exam papers which I'm having trouble with. I reckon it's actually pretty easy, but I can't get my head around it, mostly because I'm badly out of practice in this area of maths.

    Here goes:
    Let R be a finite ring with identity, and suppose for all nonzero z in R, the left multiplication map Lz :R -> R, Lz(x) = zx (for all x in R), is injective.

    Deduce that R is a field.

    It should be enough to prove that every element in the ring has an inverse, since a finite division ring is a field. I can see (though not necessarily prove) that Lz is an isomorphism

    Thanks for any advice you have


Comments

  • #2
    Fremen
    Registered Users, Registered Users 2 Posts: 2,481 ✭✭✭Fremen


    My brain finally started working late last night, so I guess you can ignore this


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