Exercise 9, § 2

Give an example of two filtered rings A, B whose filtrations are exhaustive and separated and a non-surjective homomorphism $u : A \to B$ compatible with the filtrations and such that $\mathrm{gr}(u)$ is bijective. From this deduce a counter-example to Corollary 1 to Proposition 12 when the ring A is not assumed to be complete and a counter-example to Proposition 13 when E is no longer assumed to be finitely generated (use Algebra, Chapter VIII, § 7, Exercise 3 (b)).