A Question about Differential Ideals

This paper is an extract of a paper that will soon appear in Communications in Algebra. The paper investigates the converse to a theorem which states that a differential domain D finitely generated over a field F of characteristic 0 with no nonzero proper differential ideals will have constants of differentiation C only in the field in question. The converse is known to be false but the question of whether the differential domain D can be finitely extended within its quotient field E to a domain with no nonzero proper differential ideals was raised in [M]. The paper about to appear contains two examples one of which contains a fatal error. This extract contains one example to show that the answer to the question posed is no. It is shown to be true under hypotheses which include restrictions on F and limiting the number of generators of D over C to two or the transcendence degree of D over C to one.