Definition 2

A right A-module E is called flat if it has the equivalent properties of Proposition 1.

Flat left A-modules are defined similarly. To say that a right A-module E is flat equivalent to saying that E, considered as a left $A^0$-module, is flat.

Remarks

(1) By Lemma 3 of no. 2, for a right A-module E to be flat, it is necessary and sufficient that, for every finitely generated left ideal $a$ of A, the canonical mapping $E \otimes_A a \to E$ (Proposition 1) with image Ea be injective.

(2) Let E be a flat right A-module. If $M'$ is a submodule of a left A-module M, the canonical injection $E \otimes_A M' \to E \otimes_A M$ allows us to identify $E \otimes_A M'$ with a subgroup of $E \otimes_A M$. This being so, let N be a left A-module, $u : M \to N$ a homomorphism, $K = \mathrm{Ker}\, u$, and $I = \mathrm{Im}\, u$. By considering the exact sequence

it is easily seen (Proposition 1) that $E \otimes_A (\mathrm{Ker}\, u)$ is identified with $\mathrm{Ker}(1_E \otimes u)$. On the other hand, writing $u'$ for the surjective homomorphism $M \to I$ with the same graph as $u$, and $i$ for the canonical injection $I \to N$, $1_E \otimes u'$ is surjective (no. 1, Lemma 1) and $1_E \otimes i$ is injective since E is flat. As

$E \otimes_A (\mathrm{Im}\, u)$ is identified with $\mathrm{Im}(1_E \otimes u)$.