Exercise 4, § 2

Let M be an A-module and $p$ and $q$ two projectors of M.

a) We have $p(M)\subset q(M)$ if and only if we have $qp=p$; the kernel Ker($p$) is contained in Ker($q$) if and only if we have $qp=q$.

b) The sum $p+q$ is a projector if and only if we have $pq=-qp$. If $p+q$ is a projector and if in M, the relation $2x= 0$ implies $x= 0$, then we have $pq= 0$ and $qp= 0$.

c) Prove that the following properties are equivalent:

(i) We have $pq=qp$.

(ii) We have $q(p(M))\subset p(M)$ and $q$(Ker($p$))$\subset$ Ker($p$).

(iii) There exists a sequence $(p_1, p_2, p_3, p_4)$ of pairwise orthogonal projectors of M such that $p=p_1+p_2,q=p_1+p_3$, and $1_M=p_1+p_2+p_3+p_4$.

Such a sequence, when it exists, is unique: we have $p_1=pq$.