Proposition 8

Let A, B be two rings and $\rho$ a homomorphism from A to B. Suppose that there exists a right B-module E such that $\rho_*(E)$ is a faithfully flat A-module. Then: (i) For every left A-module F, the canonical homomorphism $j : F \to F_{(B)} = B \otimes_A F$ (such that $j(x) = 1 \otimes x$ for all $x \in F$) is injective. (ii) For every left ideal a of A, $\overline{\rho}^{-1}(Ba) = a$. (iii) The homomorphisms $\rho$ is injective. (iv) For every maximal left ideal m of A, there exists a maximal left ideal n of B such that $\overline{\rho}^{-1}(n) = m$.

We prove (i). We know (Algebra, Chapter 11, § 5, no. 2, Corollary to Proposition 5) that for every right B-module M, the canonical A-homomorphism $i : M \to \rho_*(M) \otimes_A B = \rho^*(\rho_*(M))$ defined by $i(y) = y \otimes 1$ is injective and that the A-module $i(M)$ is a direct factor of $\rho_*(M) \otimes_A B$. Hence, for every left A-module F,

is injective (§ 2, no. 1, Lemma 2). Taking $M = E$, it follows (since $i \otimes 1_F = 1_M \otimes j$) that $j$ is injective (no. 1, Proposition 2).

Assertion (ii) follows from (i) by taking $F = A_s/a$ and (iii) from (ii) by taking $a = \{0\}$.

Finally, if m is a maximal left ideal of A, then $\overline{\rho}^{-1}(Bm) = m$ by (ii), and consequently $Bm \# B$. Then there exists a maximal left ideal n of B containing Bm (Algebra, Chapter I, § 8, no. 7, Theorem 2); then $m \subset \overline{\rho}^{-1}(n)$ and as $\rho(1) \notin n$, $1 \notin \overline{\rho}^{-1}(n)$. Consequently $\overline{\rho}^{-1}(n) = m$.

If $A$ and $B$ satisfy the conditions of Proposition 8, $A$ is usually identified with a subring of $B$ by means of $\rho$.