Exercise 16, § 1

a) Let A be a commutative ring and $\rho : A\rightarrow B$ an injective ring homomorphism that makes B into a finitely generated left A-module. Suppose that for every increasing (resp. decreasing) sequence $(\mathfrak{a}_n)_{n\in\mathbf{N}}$ of ideals of A, the sequence $(\mathfrak{a}_nB)_{n\in\mathbf{N}}$ is stationary (which is the case if the ring B is right Noetherian (resp. right Artinian)). Prove that the ring A is Noetherian (resp. Artinian) (use Exercise 15).

b) Give an example of a left and right Artinian and Noetherian ring B and a subring A of B such that B is finitely generated as a left and right A-module and that the ring A is neither left nor right Noetherian or Artinian (choose a suitable integral domain C, with field of fractions K, take B to be the matrix ring $\mathbf{M}_3(K)$ and A to be the subring of B consisting of the matrices $(a_{ij})_{1\leqslant i,j\leqslant 3}$ with $a_{ij}= 0$ for $1\leqslant i < j\leqslant 3$ and $a_{22}\in C.$)