Pure submodule

in the sense of Cohn

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

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

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

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

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