Chapter I(*)

Flat Modules

Unless otherwise stated, all the rings considered in this chapter are assumed to have a unit element; all the ring homomorphisms are assumed to map unit element to unit element. A subring of a ring $A$ means a subring containing the unit element of $A$.

If $A$ is a ring, $M$ a left $A$-module and $U$ (resp. $V$) an additive subgroup of $A$ (resp. $M$), recall that $UV$ or $U . V$ is used to denote the additive subgroup of $M$ generated by the products $uv$, where $u \in U, v \in V$ (Algebra, Chapter VIII, § 6, no. 1). If $a$ is an ideal of $A$, we write $a^0 = A$. For any set $E$, $1$, (or 1 if no confusion arises) is used to denote the identity mapping of $E$ to itself.

Recall that the axioms for a module imply that if $E$ is a left (resp. right) module over a ring $A$ and $1$ denotes the unit element of $A$, then $1.x = x$ (resp. $x.1 = x$) for all $x \in E$ (Algebra, Chapter II, §1, no. 1). If $E$ and $F$ are two left (resp. right) $A$-modules, recall that $\mathrm{Hom}_A(E, F)$ (or simply $\mathrm{Hom}(E, F)$) is used to denote the additive group of homomorphisms of $E$ to $F$ (loc. cit., § 1, no. 2). By an abuse of notation $O$ will often be used to denote a module reduced to its identity element.