  This article investigates the splitting problem for finitely generated
projective modules $P$ over polynomial algebras $A[T]$ on various base rings,
where $\text{rank}(P) = \dim(A)$. Our main approaches are (1) in terms of
\emph{generic sections}, and (2) in terms of \emph{monic inversion principles}.
We prove that if $P$ has a complete intersection generic section, then it
splits off a free summand of rank one, where $A$ is an affine algebra over an
algebraically closed field of characteristic $\neq 2$. We give a partial answer
to an old question due to Roitman on monic inversion principle for projective
modules over affine $\mathbb{Z}$-algebras. Whenever $A$ is an affine algebra
over $\overline{\mathbb{F}}_p$, we prove a monic inversion principle for
ideals. We further exhibit some applications.

Commutative Algebra