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Anticommuting Ring that only has left or right inverse?

  1. Nov 29, 2007 #1
    as the title says i´m looking for an example of a ring witch has only one sided inverses e.g left or right sided ones -> a*b=1 does not imply b*a =1!

    thanks for your advice :)
  2. jcsd
  3. Nov 30, 2007 #2
  4. Nov 30, 2007 #3
    cool thanks :)
    Do you know if there is also one with 0 and 1 ?
  5. Nov 30, 2007 #4
    Sorry, my mistake. What I've posted is an example of ring with only one-sided unity :-(
    I've tried UTFG, here's an example from the book Tsit-Yuen Lam: A First Course in Noncommutative Rings

    Many rings satisfying some form of "finiteness conditions" can be shown to be Dedekind-finite, but there do exist non-Dedekind-finite rings. For instance, let V be the k-vector space [tex]ke_1\oplus ke_2\oplus\ldots[/tex] with a countably infinite basis [tex]\{e_i; i\ge1\}[/tex] over the field k, and let [tex]R=\textrm{End}_k V[/tex] be the k-algebra of all vector space endomorphisms of V. If [tex]a,b\in R[/tex] are defined on the basis by
    [tex]$b(e_i)=e_{i+1}$[/tex] for all [tex]i\ge1[/tex] and
    [tex]$a(e_1)=0, a(e_i)=e_{i-1}$[/tex] for all [tex]i\ge2[/tex],
    the clearly [tex]ab=1\ne ba[/tex] so [tex]a[/tex] is right-invertible without being eft-invertible, and R gives an example of a non-Dedekind-finite ring.
    Last edited: Nov 30, 2007
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