Exercise 28, § 1

Let B be a ring, A a subring of B, and $x$ an element of B such that we have $A + Ax= A +xA$ and that the ring B is generated by $A\cup \{x\}$.

a) Let M be a Noetherian left A-module; prove that the B-module $M_{(B)}= B\otimes_AM$ is Noetherian. In particular, if the ring A is left Noetherian, then so is B.

b) Suppose that the A-module M has finite length $n$; prove that every B-submodule of $M_{(B)}$ admits a generating subset of cardinality $\leqslant n$ (adapt the proof of Theorem 1 of VIII, p. 6).