Pure submodule

in the sense of Cohn

A submodule $A$ of a right $R$- module $B$ such that for any left $R$- module $C$ the natural homomorphism of Abelian groups

$$A \otimes _ {R} C \rightarrow B \otimes _ {R} C$$

is injective. This is equivalent to the following condition: If the system of equations

$$\sum _ { i= } 1 ^ { n } x _ {i} \lambda _ {ij} = a _ {j} ,\ \ 1 \leq j \leq m ,\ \ \lambda _ {ij} \in R ,\ a _ {j} \in A ,$$

has a solution in $B$, then it has a solution in $A$( cf. Flat module). Any direct summand is a pure submodule. All submodules of a right $R$- module are pure if and only if $R$ is a regular ring (in the sense of von Neumann).

In the case of Abelian groups (that is, $R = \mathbf Z$), the following assertions are equivalent: 1) $A$ is a pure (or serving) subgroup of $B$( cf. Pure subgroup); 2) $n A = A \cap n B$ for every natural number $n$; 3) $A / n A$ is a direct summand of $B / n A$ for every natural number $n$; 4) if $C \subseteq A$ and $A / C$ is a finitely-generated group, then $A/C$ is a direct summand of $B/C$; 5) every residue class in the quotient group $B / A$ contains an element of the same order as the residue class; and 6) if $A \subseteq C \subseteq B$ and $C / A$ is finitely generated, then $A$ is a direct summand of $C$. If property 2) is required to hold only for prime numbers $n$, then $A$ is called a weakly-pure subgroup.

The axiomatic approach to the notion of purity is based on the consideration of a class of monomorphism $\mathfrak K _ \omega$ subject to the following conditions (here $A \subseteq _ \omega B$ means that $A$ is a submodule of $B$ and that the natural imbedding belongs to $\mathfrak K _ \omega$): P0') if $A$ is a direct summand of $B$, then $A \subseteq _ \omega B$; P1') if $A \subseteq _ \omega B$ and $B \subseteq _ \omega C$, then $A \subseteq _ \omega C$; P2') if $A \subseteq B \subseteq C$ and $A \subseteq _ \omega C$, then $A \subseteq _ \omega B$; P3') if $A \subseteq _ \omega B$ and $K \subseteq A$, then $A / K \subseteq _ \omega B / K$; and P4') if $K \subseteq B$, $K \subseteq _ \omega B$ and $A / K \subseteq _ \omega B / K$, then $A \subseteq _ \omega B$. Taking the class $\mathfrak K _ \omega$ instead of the class of all monomorphisms leads to relative homological algebra. For example, a module $Q$ is called $\omega$- injective if $A \subseteq _ \omega B$ implies that any homomorphism from $A$ into $Q$ can be extended to a homomorphism from $B$ into $Q$( cf. Injective module). A pure injective Abelian group is called algebraically compact. The following conditions on an Abelian group $Q$ are equivalent: $\alpha$) $Q$ is algebraically compact; $\beta$) $Q$ splits as a direct summand of any group that contains it as a pure subgroup; $\gamma$) $Q$ is a direct summand of a group that admits a compact topology; and $\delta$) a system of equations over $Q$ is solvable if every finite subsystem of it is solvable.

References

[1]	A.P. Mishina, L.A. Skornyakov, "Abelian groups and modules" , Amer. Math. Soc. (1976) (Translated from Russian)
[2]	E.G. Sklyarenko, "Relative homological algebra in categories of modules" Russian Math. Surveys , 33 : 3 (1978) pp. 97–137 Uspekhi Mat. Nauk , 33 : 3 (1978) pp. 85–120
[3]	C. Faith, "Algebra: rings, modules, and categories" , 1 , Springer (1973)
[4]	L. Fuchs, "Infinite abelian groups" , 1–2 , Acad. Press (1970–1973)

References

[a1]	J. Rotman, "Introduction to homological algebra" , Acad. Press (1979)