Seminario: “On the prime spectrum of rings of functions”

Giovedì 1 Marzo 2018, ore 15:00 - Aula 2BC30 - Sophie Frisch


Giovedì 1 Marzo 2018 alle ore 15:00 in Aula 2BC30, Sophie Frisch (TU Graz) terrà un seminario dal titolo “On the prime spectrum of rings of functions”.

Our motivation is Chabert's result that the maximal ideals of the ring of integer-valued functions on a domain D are in one to one correspondence to the elements of the M-adic completions of D. We obtain similar results for more general kinds of rings of functions by replacing elements of completions by ultrafilters on D. By a ring R of functions we mean a subring of an infinite product of copies of a domain D. The elements of R can be interpreted as functions from the index set to D. The prime ideals corresponding to ultrafilters can be seen as inverse images of prime ideals of an ultrapowers of D. We are working on finding a notion of density for R in the infinite product of copies of D that guarantees "lying over", meaning that all prime ideals of R are contractions of prime ideals of the product. This question is related to interesting properties that R may or may not have, such as the possibility to interpolate arbitrary functions at any finitely many arguments (equvalent to topological density) and also the Skolem properties relating ideals of functions to their ideals of values at each argument.