Prime Elements and Prime Sequences in Polynomial Rings

Abstract

Abstract. The central question of this note concerns the existence of prime elements in polynomial rings. In it are established for polynomial rings over arbitrary noetherian rings-insofar as is generally possible-certain results concerning bases for maximal ideals, well known for polynomial rings over fields and principal ideal domains. These results may be interpreted geometrically as theorems about normal (and especially smooth) closed points on ruled schemes.

A Negative Answer to the Prime Sequence Question

“…Reviews 54 #5206]. It is discussed in [1] and it is shown to have an affirmative answer for certain types of rings. However, the prevailing belief was that the general case would have a negative answer and here we shall demonstrate precisely that.…”