The finiteness of 𝐼 when 𝑅[𝑋]/𝐼 is 𝑅-projective

Abstract

This paper is concerned with the relationship between R[X]¡I being a projective .R-module and / being a finitely generated ideal of R[X\. It is shown that if R[X]¡I is /{-free, then I=fR[X],fa. monic polynomial of R[X]. Also, R[X]/I is a finitely generated projective A-module if and only if R[X]/I is a finitely generated .R-moduIe and I=fR[X] for some f e R[X]. When R[X]jI is projective, / is a finitely generated ideal if and only if / is a principal ideal. Finally, an example is given to show that R[X]/I can be projective without / being finitely generated.