A short exact sequence
0 → K → F → E → 0 of left modules over a ring A is l-pure if aK = K ∩aF for all
a ∈ A, and pure if for any right A-module M, the map M ⊗K → M ⊗F is injective.
A module E is torsion free (Iiattori) if its presence on the right forces l-purity, and
flat if it forces purity. Similarly, we have on the left the notions of divisibility
(Hattori) and absolute purity. Considering the functor E → En taking A-modules to
modules over the matrix ring Mn(A), a sequence is called n-pure if its image under
this functor is 1-pure; n-torsion free and n-divisible modules are similarly defined. It
is shown that purity, flatness, and absolute purity, respectively, are equivalent to
the requirement that n-purity, n-torsion-freeness, and n-divisibility should
hold for all n. n-divisibility and absolute purity are preserved under direct
sums, products and certain inductive limits; n-torsion-freeness and flatness
under direct sums and inductive limits, but not products. A condition is
given guaranteeing that products of at most a given cardinality preserve
n-torsion-freeness. It is shown that if every left ideal of A is generated by
at most n elements, then n-torsion-freeness is equivalent to flatness. The
behavior of these properties under localization is studied, and it is shown that if
A is locally a domain then the two notions of purity agree if and only if
w. gl. dim. (A) ≦ 1.
