Homological properties of the ring of differential polynomials

Abstract

The ring of differential polynomials over a universal differential field (Kolchin [7]), and the ring of twisted polynomials F 2 [), p], where F2 is an algebraic closure of Z/2Z and p is the automorphism of JF 2 defined by: &->z 2 , "localized" at the multiplicative subset {t k \k an integer^0}, provide examples of a principal right and left ideal domain R, not a field, that is a right F-ring (i.e., each simple right i-module is injective). Such a ring was conjectured to exist by Carl Faith. Both examples are shown to have a unique simple right -R-module. If R is either example, then by definition of a right F-ring, every right i?-module has a maximal submodule. Bass proved that if a ring A satisfies the d.c.c. on principal left ideals, then A has a bounded number of orthogonal idempotents and every right Amodule has a maximal submodule. The above examples show that the converse is false, thus answering a question raised by Bass [l, p. 470].