I am reposting this as earlier question was probably not clear to some.
So I was reading a book in which we have a commutative ring $A$, a commutative algebra over it called $B$ and a set of projective modules over $B$, say $X_i$. The book has following statement
"the modules $X_i$ are projective, hence become free (of rank 1) over a suitable ring $C$ containing $B$."
This statement was only used and not proved. I have never seen something similar so I don't know how and why is it true. I simply tried googling it but I couldn't find anything. If someone here has seen a similar result and they can either explain it or lead me to a reference text then I will be grateful.
Edit : for the (rank 1) part, it is possible that this is true only in given situation and not in general. For bit more context the modules $X_i$ are $e_iB^d$ where $d$ is some positive integer and $\{e_i\}$ form a finite family of orthogonal idempotents of $B$ whose sum is identity of $B$.