Exercise 6, § 2

Let A be a ring and $u$ and $v$ two elements of A such that $uv= 1$ and $vu\not= 1$. For $i\geqslant 1$ and $j\geqslant 1$, we set $e_{ij}=v^{i-1}u^{j-1}-v^iu^j$.

a) Prove that we have $e_{ij}\not= 0$ and $e_{ij}e_{hk}=\delta_{jh}e_{ik}$ for $i, j, k, h$ integers $\geqslant 1$.

b) Prove that the left ideals $Ae_{ii}$ of A (for $i\geqslant 1$) are nonzero, pairwise isomorphic as A-modules (use Exercise 5 d)), and that their sum is direct.

c) Prove that we have $u(v+e_{1i}) = 1$ for every $i\geqslant 1$ and $e_{1i}\not=e_{1j}$ for $i\geqslant 1,j\geqslant 1$, $i\not=j$.

d) Let B be a ring that has the equivalent properties of Exercise 7 of VIII, p. 15, a). Deduce from the above that every left or right invertible element of B is invertible.