## Modules where every element is contained in a linearly independent set of a given cardinality

Let $R$ be a commutative ring and let $M$ be a $R$-module such that every element $m\in M$ is contained in a set of linearly independent elements of cardinality $n$. How could one characterize such modules? Are there results about when a module is of the above type? In particular, when is it true for finitely generated projective modules of constant rank $\geq n$?

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