Exercise 3, Appendix 1

Let A be a $k$-algebra, $\mathfrak{a}$ a regular left ideal of A, and $u$ and $v$ right units modulo $\mathfrak{a}$. a) Give an example where $u-v$ does not belong to $\mathfrak{a}$ (cf. III, §9, p. 644, Exercise 1). Deduce an example where the mapping $\widetilde{\mathfrak{a}}\mapsto \widetilde{\mathfrak{a}}\cap A$ (Proposition 3 of VIII, p. 437) is not injective.

b) Suppose that the ideal $\mathfrak{a}$ is two-sided and that the $k$-algebra $B = A/\mathfrak{a}$ does not contain any nonzero two-sided ideal $\mathfrak{b}$ such that $B\mathfrak{b}= 0$. Prove that $u-v$ belongs to $\mathfrak{a}$. This is, in particular, the case when A is commutative.