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Quotient fields of residue class rings of function rings

1960; Duke University Press; Volume: 4; Issue: 3 Linguagem: Inglês

10.1215/ijm/1255456059

1945-6581

Leonard Gillman, Meyer Jerison,

Algebraic structures and combinatorial models

The ring of all continuous functions from a topological space X into the reals, R, is denoted by C or C(X).In [1, 14E], an example is presented of a residue class field of one ring of continuous functions that is isomorphic in a natural way with the quotient field of a residue class ring of another ring of functions.The first ring is C(II), ll denoting the discrete space of positive integers.The second is C(2), where 2] ll u z} is the subspace of the Stone-Cech compactifiction of ll obtained by adioining a single point z to ll.The set M of all functions in C(II) that vanish on set having z in its closure is a maximal ideal in C(II) the set Q of all functions in C(2) that vanish on a neighborhood of z is a nonmaximal prime ideal in C(2).In the manner to be described in 2, the mpping that sends each function in C(2;) into its restriction to ll induces an isomorphism of the integral domain C(Z)/Q onto a subring of C(II)/M, and C(II)/M is the quotient field of that subring.In the present pper, we investigate the possibility of obtaining, in a similar way, the quotient field of C(Y)/Q, where Q is a prime ideal in an arbitrary function ring C(Y).We shall find that a necessary condition is that Q be z-ideal, i.e., if h C(Y), and if there exists g e Q such that h(y) 0 wherever g(y)O, then h e Q.A sufficient condition is that Q have an immediate successor in the family of all z-ideals in C(Y).On the other hand, if Q is the intersection of countable family of z-ideals different from itself, then the quotient field of C(Y)/Q is not isomorphic with a residue class field of any function ring.The question is left open as to what may hppen in case Q neither has an immediate successor nor is a countable intersection; whether such prime z-ideal Q exists at all is lso left unsettled. PreliminariesThe terminology and notation of [1] will be used throughout the paper.In this section, we summarize the material from [1] that will be used.Most of the information about prime ideals can also be found in [2] and [3].When deling with algebraic properties of a ring C(X), one loses no gen- erality by supposing X to be completely regular.We adopt this standing assumption.