# Anticommuting Ring that only has left or right inverse?

1. Nov 29, 2007

### 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!

2. Nov 30, 2007

### kompik

3. Nov 30, 2007

### Mr.Brown

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

4. Nov 30, 2007

### kompik

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.

Last edited: Nov 30, 2007