Exercise 24, § 2

A submodule $M'$ of a right $A$-module $M$ is called pure if, denoting by $j : M' \to M$ the canonical injection, the homomorphism

is injective for every left $A$-module $N$. This is so if $M'$ is a direct factor of $M$ or if $M/M'$ is flat, but these two conditions are not necessary (Exercise 4).

(a) Show that, for $M'$ to be a pure submodule of $M$, it is necessary and sufficient that, if $(m'_i)_{i \in I}$ is a finite family of elements of $M'$, $(x_j)_{j \in J}$ a family of elements of $M$ such that $m'_i = \sum_{j \in J} x_j a_{ji}$ for all $i \in I$ and a family $(a_{ji})$ of elements of $A$, then there exists a family $(x'_j)_{j \in J}$ of elements of $M'$ such that $m'_i = \sum_{j \in J} x'_j a_{ji}$ for all $i \in I$. (To see that the condition is sufficient, use Lemma 10 of no. 11 to show that $M' \otimes_A N \to M \otimes_A N$ is injective for every finitely generated left $A$-module $N$; to see that the condition is necessary, consider a finitely generated left $A$-module $N = L/R$, where $L$ is a finitely generated free $A$-module and $R$ is a finitely generated submodule of $L$.) Deduce from this criterion that, if $A$ is a principal ideal domain, the notion of pure submodule of an $A$-module coincides with that of Algebra, Chapter VII, § 2, Exercise 7.

(b) Let $M$ be a right $A$-module, $M'$ a submodule of $M$ and $M''$ a submodule of $M'$. Show that, if $M'$ is a pure submodule of $M$ and $M''$ a pure submodule of $M$, then $M''$ is a pure submodule of $M$ and $M'/M''$ is a pure submodule of $M/M''$. If $M''$ is a pure submodule of $M$, $M''$ is a pure submodule of $M'$.

(c) Show that, if $N$ and $P$ are two submodules of $M$ such that $N \cap P$ and $N + P$ are pure in $M$, then $N$ and $P$ are pure submodules of $M$. Give an example of two submodules $N, P$ of $\mathbf{Z}^2$ which are pure in $\mathbf{Z}^2$ but where $N + P$ is not a pure submodule of $\mathbf{Z}^2$.

(d) Let $C$ be a commutative ring and $E, F$ two $C$-modules; show that, if $E'$ (resp. $F'$) is a pure submodule of $E$ (resp. $F$), the canonical mapping $E' \otimes_C F' \to E \otimes_C F$ is injective and identifies $E' \otimes_C F'$ with a pure submodule of $E \otimes_C F$.

(e) Let $\rho : A \to B$ be a ring homomorphism, $M$ a right $A$-module and $M'$ a pure submodule of $M$. Show that $M'_{(B)} = M' \otimes_A B$ is canonically identified with a pure submodule of $M_{(B)} = M \otimes_A B$.