Vector spaces over a field (mathematics) are flat modules.

A -module is called a flat module if the tensor product functor is exact.

He began his academic career by working in commutative algebra, especially on flat modules.

As a consequence, one can deduce that every finitely-presented flat module is projective.

The completion of a Noetherian ring R is a flat module over R.

Some crossover occurs for projective or flat modules.

See flat module or, for more generality, flat morphism.

From this one can deduce that pure submodules of flat modules are flat.

The product of the local rings of a commutative ring is a faithfully flat module.

In contrast, no choice is needed to prove that free modules are flat, so theorems about flat modules can still apply.