Exercise 7, § 2

(a) With the notation of Exercise 6, show that, if M is a projective module, then $\lambda(M) = -1$ or $\lambda(M) = +\infty$. If A is a left Noetherian ring, then, for every A-module M, $\lambda(M) = -1$ or $\lambda(M) = +\infty$.

(b) If $a$ is a left ideal of a ring $A$, which is not finitely generated, $A_s/a$ is a monogenous $A$-module which does not admit a finite presentation, in other words $\lambda(A_s/a) = 0$ (no. 8, Lemma 9).

(c) Give an example of a monogenous left ideal $a$ of a ring $A$ such that $A_s/a$ (which has a finite presentation) admits a dual which is not a finitely generated right $A$-module.

(d) Let $K$ be a commutative field, $E$ the vector space $K^{(\mathbf{N})}$, $(e_n)$ the canonical basis of $E$ and $T$ the tensor algebra of $E$, which therefore has a basis consisting of the finite products $e_{i_1}e_{i_2}\ldots e_{i_k}$ ($k \geqslant 0, i_j \in \mathbf{N}$ for all $j$). For a given integer $n$, let $b$ be the two-sided ideal of $T$ generated by the products

and $e_{n+k}e_n$ for all $k \geqslant 1$; let $A$ be the quotient ring $T/b$, and for every integer $m$, let $a_m$ be the canonical image of $e_m$ in $A$. Show that, if $M = A_s/Aa_0$, then $\lambda(M) = n$ (observethat, for $m \leqslant n - 1$, the left annihilator of $a$, is $Aa_{m+}$, and use Exercise 6(b)).