Exercise 13, § 1

a) Let $\rho : A\rightarrow B$ be a ring homomorphism. Assume that A is left Noetherian and that there exists a finite subset S of B whose elements commute with one another and with the elements of $\rho$(A), such that the ring B is generated by $\rho (A)$ and S. Then B is left Noetherian.

b) Let A be a nonzero commutative ring. Give an example of an associative unital A-algebra, generated by finitely many elements, that is a neither left nor right Noetherian ring (consider the free associative algebra Libas$_A(I)$ (III, §2, No. 7, p. 449, Definition 2), where I is a set of cardinality $\geqslant 2$).