Exercise 9, § 2

Given an A-module E, for every submodule M of E (resp. every submodule M’ of E*), we denote by $M^0$ (resp. ${M'}^0$) the orthogonal of M in E* (resp. the orthogonal of M’ in E). We consider the following four conditions: (A) The canonical homomorphism $c_E : E \to E^{**}$ is bijective. (B) For every submodule M of E, the canonical homomorphism $E^*/M^0 \to M^*$ is bijective. (C) For every submodule M of E, one has $M^{00} = M$. (D) For every pair of submodules M, N of E, one has

a) Show that condition (B) implies (D) (prove that for $x^* \in (M \cap N)^0$, 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 without zero divisor, but having no left fractions field (for example the tensor algebra of a vector space of dimension > 1, cf. III, p. 185, exerc. 5). If we take $E = A_s$, condition (A) is satisfied, but none of the conditions (B), (C), (D) (for an example where (B), (C), (D) are verified, but not (A), see II, p. 103, th. 6). c) We take for A 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 not reduced to 0; then show that condition (B) implies (C). e) The kernel of $c_E$ is the orthogonal $(E^*)^0$ of $E^*$ in E. Give an example where neither $E^*$ nor $(E^*)^0$ are reduced to 0 (consider a module containing an element whose annihilator contains a non-zero divisor element). f) Let M be a direct factor of E; show that one has $M^{00} = M + (E^*)^0$.