Exercise 12, § 1

Let K be a commutative field. Denote by $(e_n)_{n\in\mathbf{N}}$ the canonical basis of the vector space $V = K^{(\mathbf{N})}$. We define the sequence $(u_n)_{n\in\mathbf{N}}$ of endomorphisms of V by setting

$u_0(e_0) = 0u_0(e_1) = 0u_0(e_{n+1}) =e_n$ for $n\geqslant 1$

$u_m(e_0) =e_mu_m(e_n) = 0$ for $m\geqslant 1$ and $n\geqslant 1$.

Let A be the unital K-subalgebra of End$_K(V)$ generated by the sequence $(u_n)_{n\in\mathbf{N}}$. Prove that the left A-module V is Artinian, finitely generated (and even monogenous), but not Noetherian.