If R is a commutative ring and P is a projective R-module, then a submodule B of P is said to be basic if the image of B in P/mP is nonzero for every maximal ideal m of R, or equivalently, if locally at each maximal ideal of R, B contains a nontrivial direct summand of P. The ring R is said to be a bcs ring if the following equivalent conditions hold:
(i) every basic submodule of a projective R-module contains a rank one summand; \((ii)\quad every\) finitely generated basic submodule of a projective R-module contains a rank one summand; \((iii)\quad every\) basic submodule of a finitely generated projective R-module P contains a rank one summand of P; \((iv)\quad for\) every n, every finitely generated basic submodule of \(R^ n\) contains a rank one summand of \(R^ n.\)
The authors mention that motivation for the study of bcs rings comes from the fact that such rings are pole assignable. The contour of the class of bcs rings is very nicely sketched. It is shown that if R is a bcs ring, then so also is every quotient R/I, and the map Pic(R)\(\to Pic(R/I)\) is surjective for every ideal I of R. The authors prove that 0-dimensional rings, semilocal rings, 1-dimensional domains, and 1-dimensional Noetherian rings are bcs rings. If V is a semilocal principal ideal domain, it is shown that the polynomial ring V[x] is a bcs ring. On the other hand, the authors mention that they do not know of any 2- dimensional affine algebras which are bcs rings. They prove that if R is an affine algebra of dimension \(\geq 2\) over a field and if \(Pic(R)=0\), then R is not a bcs ring.