Editing Flat module (section)

=== Direct sums, limits and products ===
A [[direct sum of modules|direct sum]] <math>\textstyle\bigoplus_{i \in I} M_i</math> of modules is flat if and only if each <math>M_i</math> is flat.

A [[direct limit]] of flat is flat. In particular, a direct limit of [[free module]]s is flat. Conversely, every flat module can be written as a direct limit of [[finitely generated module|finitely-generated]] free modules.{{sfn|Lazard|1969|ps=none}}

[[Direct product]]s of flat modules need not in general be flat. In fact, given a ring {{mvar|R}}, every direct product of flat {{mvar|R}}-modules is flat if and only if {{mvar|R}} is a [[coherent ring]] (that is, every finitely generated ideal is finitely presented).{{sfn|Chase|1960|ps=none}}