Given a subset of linearly indept elements of a vector space $V$, one can always extend to a basis $B$ of $V$. Is there a corresponding result for a projective module $P$ over a unital ring $R$? In other words, when can a subset $S \subseteq P$ be extended to a dual basis of $P$? Is it always possible when $S$ consists of a single element?
To clarify - by "dual basis" I mean the usual equivalent formulation of projectivity in terms of a dual basis, as explained for example here:
No. Let $R=\mathbb{Z}$. Then $2$ cannot be extended to a basis of $\mathbb{Z}$ (which is free, hence projective).
If you want an example in higher dimension, take $(2,2)\in\mathbb{Z}^2$. It cannot be extended to a basis of $\mathbb{Z}^2$ (the reason is that the coordinates of $(2,2)$ are not coprime).