Anticommuting Ring that only has left or right inverse?

[Mr.Brown]

Hi
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 :)

 

[kompik]

Hi
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 :)

What about this http://planetmath.org/encyclopedia/KleinFourRing.html"

 

[Mr.Brown]

cool thanks :)
Do you know if there is also one with 0 and 1 ?

 

[kompik]

cool thanks :)
Do you know if there is also one with 0 and 1 ?

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 ke_1\oplus ke_2\oplus\ldots with a countably infinite basis \{e_i; i\ge1\} over the field k, and let R=\textrm{End}_k V be the k-algebra of all vector space endomorphisms of V. If a,b\in R are defined on the basis by
$b(e_i)=e_{i+1}$ for all i\ge1 and
$a(e_1)=0, a(e_i)=e_{i-1}$ for all i\ge2,
the clearly ab=1\ne ba so a is right-invertible without being eft-invertible, and R gives an example of a non-Dedekind-finite ring.