Rings of Integer-Valued Polynomials and the bcs-Property

For a commutative ring R, recall that a submodule B of a projective module P is called basic if https://www.w3.org/1998/Math/MathML"> B M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429222641/bd60306a-e05b-46de-85c8-902aafc6d56a/content/eq557.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> contains a nontrivial summand of https://www.w3.org/1998/Math/MathML"> P M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429222641/bd60306a-e05b-46de-85c8-902aafc6d56a/content/eq558.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , (or, equivalently, if the image of B in https://www.w3.org/1998/Math/MathML"> P / M P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429222641/bd60306a-e05b-46de-85c8-902aafc6d56a/content/eq559.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is nonzero), for every maximal ideal https://www.w3.org/1998/Math/MathML"> M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429222641/bd60306a-e05b-46de-85c8-902aafc6d56a/content/eq560.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> of R. A commutative ring R is called a bcs-ring if and only if for every positive integer n, every finitely generated basic submodule of R ⁿ contains a rank one projective summand of R ⁿ . (This notion was introduced by M. Hautus and E. Sontag in [14] and called “property (†).” In [1], J. Brewer, D. Katz, and W. Ullery referred it to as the “UCS-property” It was called the “bcs-property” and considered systematically by C. Weibel and W. Vasconcelos in [15].) For over a decade, the present authors have been interested in knowing whether all Prüfer domains are bcs-rings. This question is analogous to the question, “Is every Bézout domain an elementary 66divisor ring?” One place one might look for a counterexample is in “rings of integer-valued polynomials.”