Lemma 10.101.1. Let (R, \mathfrak m) be a local ring with nilpotent maximal ideal \mathfrak m. Let M be a flat R-module. If A is a set and x_\alpha \in M, \alpha \in A is a collection of elements of M, then the following are equivalent:
\{ \overline{x}_\alpha \} _{\alpha \in A} forms a basis for the vector space M/\mathfrak mM over R/\mathfrak m, and
\{ x_\alpha \} _{\alpha \in A} forms a basis for M over R.
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