Exercise 7, § 3

a) With the notations of Exercise 6, now prove that, if $M$ is a projective module, one has

If $A$ is a left noetherian ring, then, for every $A$-module $M$, one has $\lambda(M) = -1$ or $\lambda(M) = +\infty$.

b) If $a$ is a left ideal of a ring $A$, which is not of finite type, $A_s/a$ is a monogeneous $A$-module which does not admit a finite presentation, in other words $\lambda(A_s/a) = 0$ (cf. X, p. 11, prop. 6).

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

d) Let $K$ be a commutative field, $E$ the vector space $K^{(N)}$, $(e_n)$ the canonical basis of $E$, $T$ the tensor algebra of $E$, whose basis is therefore formed by the finite products $e_{i_1} e_{i_2} \ldots e_{i_k}$ ($k \geqslant 0, i_j \in \mathbf{N}$ for every $j$). For a given integer $n$, let $b$ be the two-sided ideal of $T$ generated by the products $e_1 e_0, e_2 e_1, \ldots, e_n e_{n-1}$ and $e_{n+k} e_n$

EXERCISES

for every $k \geqslant 1$; let $A$ be the quotient ring $T/b$, and for every integer $m$, let $a_m$ be the canonical image of $e_m$ in $A$. Now prove that, if $M = A_s / Aa_o$, one has $\lambda(M) = n$ (observe that, for $m \leqslant n - 1$, the left annihilator of $a_m$ is $Aa_{m+1}$, and use Exercise 6, b)).