Luis F. Caceres-Duque
University of Puerto Rico
Mayaguez Campus

TITLE: Ultraproduct of Ideals in a Noetherian Ring

 For ideals or more generally congruences we introduce a propositional 
calculus whose models are precisely the ideals of a ring or congruences 
of an algebra. We have propositional versions of the usual ultraproduct 
theorem of logic which here assert that the sentences of this 
propositional logic true in an ultraproduct of ideals are exactly those 
which are true on an ultrafilter set of the factor ideals. We use a 
notion of ultraproduct of sets which coincides with taking limits in the 
Cantor space 2^R for a commutative ring with identity R. We are able to 
prove that the ascending chain condition on ideals of a ring is 
equivalent to any ultraproduct of these ideals always being equal to an 
intersection of some of these ideals over some ultrafilter set. In this 
way not only we do obtain a topological connection between these ideals 
but in a Noetherian ring results are obtained concerning certain ideals 
being equal to infinite intersections of other ones. One new result here 
concerning just the structure of prime ideals is that given infinitely 
many distinct prime ideals of a Noetherian ring then there must be an 
intersection of infinitely many of them which is again a prime ideal.