Exercise 9, § 2

Given an A-module E, for every submodule M of E (resp. every submodule $M'$ of $E^*$), let $M^\circ$ (resp. ${M'}^\circ$) denote the orthogonal of M in $E^*$ (resp. the orthogonal of $M'$ in E). Consider the following four properties: (A) The canonical homomorphism $c_E : E \to E^{**}$ is bijective. (B) For every submodule M of E, the canonical homomorphism $E^*/M^\circ \to M^*$ is bijective. (C) For every submodule M of E, $M^{\circ\circ} = M$. (D) For every ordered pair of submodules M, N of E,

(a) Show that condition (B) implies (D) (prove that for $x^* \in (M \cap N)^\circ$, there exists $y^* \in E^*$ such that $\langle x + y, y^* \rangle = \langle x, x^* \rangle$ for $x \in M$ and $y \in N$).

(b) Let A be a ring with no divisors of zero, but having no field of left fractions (for example the tensor algebra of a vector space of dimension > 1, cf. III, § 5, Exercise 5). If we take $E = A_s$, condition (A) is satisfied, but none of conditions (B), (C), (D) (for an example where (B), (C), (D) hold, but not (A), see § 7, no. 5, Theorem 6).

(c) Take A to be a product ring $K^I$, where K is a commutative field and I an infinite set. Show that the A-module $E = A$ satisfies conditions (A) and (B), but not (C) (observe that the annihilators of the ideals of A are all of the form $K^J$, where $J \subset I$).

(d) Suppose that for two submodules M, N of E such that $N \subset M$ and $N \neq M$, the dual $(M/N)^*$ is non-zero; then show that condition (B) implies (C).

(e) The kernel of $c_E$ is the orthogonal $(E^*)^\circ$ of $E^*$ in E. Give an example where neither $E^*$ nor $(E^*)^\circ$ is 0 (consider a module containing an element whose annihilator contains an element which is not a divisor of 0).

(f) Let M be a direct factor of E; show that $M^{\circ\circ} = M + (E^*)^\circ$.