Examples
- Every direct summand of M is pure in M. Consequently, every subspace of a vector space over a field is pure.
- (Lam 1999, p.154) Suppose
is a short exact sequence of R modules, then:
- C is a flat module if and only if the exact sequence is pure exact for every A and B. From this we can deduce that over a von Neumann regular ring, every submodule of every R-module is pure. This is because every module over a von Neumann regular ring is flat. The converse is also true. (Lam 1999, p.162)
- Suppose B is flat. Then the sequence is pure exact if and only if C is flat. From this one can deduce that pure submodules of flat modules are flat.
- Suppose C is flat. Then B is flat if and only if A is flat.
Read more about this topic: Pure Submodule
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