Exercise 2, Appendix 1

Let A be a $k$-algebra, and let $\mathfrak{a}$ and $\mathfrak{b}$ be regular (left) ideals of A.

a) The ideal $\mathfrak{a}\cap \mathfrak{b}$ is regular if and only if there exist right units $u$ modulo $\mathfrak{a}$ and $v$ modulo $\mathfrak{b}$ such that $u-v\in \mathfrak{a}+\mathfrak{b}$.

b) Deduce that $\mathfrak{a}\cap \mathfrak{b}$ is regular if $\mathfrak{a}$ is two-sided and right regular, or if $\mathfrak{a}$ is maximal. c) Let K be a commutative field, L be the free associative K-algebra in two variables X and Y (III, §2, No. 7, p. 448), and A be the K-subalgebra of L consisting of the elements of degree $\geqslant 1$. Prove that the ideals $A(1-X)$ and $A(1-Y)$ of A are regular but that their intersection is reduced to 0.