Proposition 26

Let $A$ be an integral domain, $K$ its field of fractions, $E$ an $A$-module and $E_{(K)} = K \otimes_A E$ the vector space over $K$ obtained by extending the ring of operators; let $\phi$ denote the canonical $A$-linear mapping $x \mapsto 1 \otimes x$ of $E$ into $E_{(K)}$.

(i) Every element of $E_{(K)}$ is of the form $\lambda^{-1} \phi(x)$ for $\lambda \in A, \lambda \neq 0$ and $x \in E$. (ii) The kernel of $\phi$ is the torsion submodule $T(E)$ of $E$.

(i) Every element of $E_{(K)}$ is of the form $z = \sum_{i=1}^n \xi_i \phi(x_i)$ with $\xi_i \in K$ and $x_i \in E$; for all $i$, there exists $\alpha_i \in A$ such that $\alpha_i \neq 0$ and $\alpha_i \xi_i \in A$; if $\alpha = \prod_{i=1}^n \alpha_i$, then $\alpha \neq 0$ and $\alpha \xi_i = \beta_i \in A$ for all $i$, whence, in $E_{(K)}$,

since $\phi$ is $A$-linear.

(ii) If $x \neq 0$ is not free in $E$, there exists $\alpha \neq 0$ in $A$ such that $\alpha x = 0$, whence $\alpha \phi(x) = \phi(\alpha x) = 0$ in $E_{(K)}$, which implies $\phi(x) = 0$. Conversely, suppose that, for some $x \in E$, $1 \otimes x = 0$ in $E_{(K)}$; we show that $x$ is a torsion element of $E$. We consider the set $\mathfrak{M}$ of monogenous sub-$A$-modules of $K$; this is a right directed set under the relation of inclusion, for any two elements $\alpha, \beta$ of $K$ can be written as $\alpha = \zeta^{-1} \xi, \beta = \zeta^{-1} \eta$, where $\xi, \eta, \zeta$ belong to $A$ and $\zeta \neq 0$, hence $A.\alpha \subset A.\zeta^{-1}$ and $A.\beta \subset A.\zeta^{-1}$. Moreover $K$ is the union of the modules $M \in \mathfrak{M}$ and can therefore be considered as the direct limit of the direct system defined by the modules $M \in \mathfrak{M}$ and the canonical injections (§ 6, no. 2, Remark). Hence also, to within a canonical isomorphism, $E_{(K)} = \lim \rightarrow (M \otimes_A E)$ (§ 6, no. 3, Proposition 7) and the relation $1 \otimes x = 0$ in $E_{(K)}$ implies that there exists an

M ∈ $\mathscr{P}$ such that l ∈ M and l ⊗ x = 0 in the tensor product M ⊗_A E (Set Theory, III, § 7, no. 5, Lemma 1). It may further be supposed (replacing if need be M by a monogenous submodule M' ⊃ M of K) that M = A.γ^{-1}, where γ ∈ A and γ ≠ 0. Now the mapping ξ ↦ γξ is an isomorphism of M onto the A-module A; on the other hand, the canonical isomorphism A ⊗_A E → E (§ 3, no. 4, Proposition 4) maps ξ ⊗ x to the element ξx of E; thus there exists an isomorphism M ⊗_A E → E which maps the tensor product ξ ⊗ x to the element (γξ)x of E. The hypothesis l ⊗ x = 0 in M ⊗_A E thus implies γx = 0.