Let R be a commutative Noetherian ring of dimension d. First, we define
the "geometric subring" A of a polynomial ring R[T] of dimension d+1 (the
definition of geometric subring is more general, see (1.2)). Then we prove that
every locally complete intersection ideal of height d+1 is a complete
intersection ideal. Thus improving the general bound of Mohan Kumar
\cite{NMK78} for an arbitrary ring of dimension d+1. Afterward, we deduce
that every finitely generated projective A-module of rank d+1 splits off a
free summand of rank one. This improves the general bound of Serre
\cite{Serre58} for an arbitrary ring. Finally, applications are given to a
set-theoretic generation of an ideal in the geometric ring A and its
polynomial extension A[X].Comment: 12 page