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