2007-2008

Mel Henriksen

**Monday, October 8
Davidson Lecture
Hall at Claremont McKenna, 4:15 PM**

**Title: **Residue class
rings of real analytic and real entire functions

Abstract: Let A(R) and E(R) denote respectively the ring of real analytic and real entire functions of one variable with the usual pointwise operations. It is shown that if M is a maximal ideal of A(R), then A(R)/M is isomorphic either to the reals or to a real-closed field that is an eta-one set, while if M is a maximal ideal of E(R), then E(R)/M isomorphic either to one or the latter two fields or the field of complex numbers. Moreover, residue class rings of prime ideals of these rings and their Krull dimensions (i.e., the supremum of the lengths of chains of prime ideals). Use is made of a classical but not well known theorem of E. Steinitz and techniques described in the book by L.Gillman and M. Jerison on Rings of Continuous Functions. Unfamiliar definitions will be supplied.