ON MAXIMAL IDEALS OF R∞L

ON MAXIMAL IDEALS OF R_∞L

Document Type : Original Manuscript

Authors

¹ Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran. Email: aaestaji@hsu.ac.ir and aaestaji@gmail.com

² Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran. Email: m.darghadam@yahoo.com

Abstract

Let

L

be a completely regular frame and

R L

be the ring of real-valued continuous functions
on

L

.
We consider the set

R_{\infty} L = {φ \in R L :↑ φ (\frac{- 1}{n}, \frac{1}{n}) is a compact frame for any n \in N} .

Suppose that

C_{\infty} (X)

is the family of all functions

f \in C (X)

for which the
set

{x \in X : | f (x) | \geq \frac{1}{n}}

is compact, for every

n \in N

.
Kohls has shown that

C_{\infty} (X)

is precisely the intersection
of all the free maximal ideals of

C^{*} (X)

.
The aim of this paper is to
extend this result to
the real continuous functions on a
frame and hence we show that

R_{\infty} L

is precisely the intersection
of all the free maximal ideals of

R^{*} L

.
This result is used to characterize the maximal ideals in

R_{\infty} L

Keywords