INTERSECTION OF ESSENTIAL IDEALS IN THE RING OF REAL-VALUED CONTINUOUS FUNCTIONS ON A FRAME

Document Type : Original Manuscript

Authors

¹ Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabze- var, Iran.

² Department of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan,

³ Esfarayen University of Technology, Esfarayen, Iran.

Abstract

A frame

L

is called {\it coz-dense} if

Σ_{c o z (α)} = \emptyset

implies

α = 0

. Let

R L

be the ring of real-valued continuous functions on a coz-dense and completely regular frame

L

. We present a description of the socle of the ring

R L

based on minimal ideals of

R L

and zero sets in pointfree topology. We show that socle of

R L

is an essential ideal in

R L

if and only if the set of isolated points of

Σ L

is dense in

Σ L

if and only if the intersection of any family of essential ideals is essential in

R L

. Besides, the counterpart of some results in the ring

C (X)

is studied for the ring

R L

. For example, an ideal

E

R L

is an essential ideal if and only if

⋂ Z [E]

is a nowhere dense subset of

Σ L .

Keywords