Maximally differential prime ideals (Q1074670)

Differential properties of prime ideals in commutative noetherian rational algebras R are examined. An ideal P of R is called maximally differential if there is a set D of derivations from R to R such that P is maximal among proper D-invariant ideals of R; any such P is prime. Among 1-dimensional local domains R, examples are constructed in which \((a)\quad 0\quad is\) a maximally differential ideal of R while the embedding dimension and the multiplicity of R are arbitrarily large, or \((b)\quad 0\quad is\) a maximally differential ideal of R while 0 is not even a prime ideal of the completion, or \((c)\quad 0\quad is\) a maximally differential ideal f the completion of R while 0 is not a maximally differential ideal of R. Examples (b) and (c) provide negative answers to a question of \textit{B. Singh} [J. Algebra 82, 331-339 (1983; Zbl 0514.13018)].