Notes on a Class of Ideals in Intermediate Rings of Continuous Functions
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Notes on a Class of Ideals in Intermediate Rings of Continuous Functions Mehdi Parsinia1 Received: 24 May 2019 / Revised: 31 August 2019 / Accepted: 9 October 2019 © Iranian Mathematical Society 2019
Abstract In this paper, we give some properties of z ◦ -ideals in intermediate rings of C(X ) (i.e., subrings of C(X ) containing C ∗ (X )). Moreover, some characterizations of almost P-spaces via intermediate rings of C(X ) are established. Using these, we investigate answers to the two questions concerning z ◦ -ideals in intermediate rings raised in Bag et al. (Appl Gen Topol 20(1): 109–117, 2019). Keywords Intermediate ring · z-Ideal · z ◦ -Ideal · Almost P-space Mathematics Subject Classification 54C30 · 46E25
1 Introduction Throughout this article, all topological spaces are assumed to be Tychonoff. For a given topological space X, C(X ) denotes the algebra of all real-valued continuous functions on X and C ∗ (X ) denotes the subalgebra of C(X ) consisting of all bounded elements. A subring A(X ) of C(X ) is called an intermediate ring, if C ∗ (X ) ⊆ A(X ). Intermediate C-rings are intermediate rings which are also isomorphic with C(Y ) for some Tychonoff space Y . The reader is referred to [1,11] for undefined terms and notations concerning C(X ) and intermediate rings of C(X ). The mapping S A from ˇ an intermediate ring A(X ) to the collection of closed subsets of β X (the Stone–Cech ∗ compactification of X ) defined by S A ( f ) = { p ∈ β X : ( f g) ( p) = 0, ∀g ∈ A(X )} has been first introduced in [19] and recently extensively used by the author of the present paper in studying intermediate rings of C(X ), where it is shown, parallel to the zero-set mapping Z in C(X ) (which assigns to each f ∈ C(X ), the zero set of
Communicated by Shirin Hejazian.
B 1
Mehdi Parsinia [email protected] Departmant of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran
123
Bulletin of the Iranian Mathematical Society
f in X ) it plays an important role in the context of intermediate rings, see [1,4,13– 16]. It could be easily observed that S A has some similar properties to the zero-set mapping Z , namely S A ( f g) = S A ( f ) ∪ S A (g), S A ( f 2 + g 2 ) = S A ( f ) ∩ S A (g) and S A ( f n ) = S A ( f ) for each f , g ∈ A(X ) and each n ∈ N. Also, SC ( f ) = clβ X Z ( f ) p for each f ∈ C(X ) and SC ∗ ( f ) = Z ( f β ) for each f ∈ C ∗ (X ). We use M A to denote p the set { f ∈ A(X ) : p ∈ S A ( f )} for each p ∈ β X . Evidently, MC = M p and p MC ∗ = M ∗ p . From [19, Theorem 2.8 and Theorem 2.9], it follows that the collection p of all the maximal ideals of an intermediate ring A(X ) is {M A : p ∈ β X }. It follows that f ∈ A(X ) is invertible in A(X ) if and only if S A ( f ) = ∅. For an ideal I in A(X ) we p designate by θ A (I ) the set f ∈I S A ( f ) which is clearly the set { p ∈ β X : I ⊆ M A }; θC (I ) is simply denoted by θ (I ) for each ideal I in C(X ). Note that by an ideal we always mean a proper ideal. The aim of this paper is to give some properties of z ◦ -ideals in intermediate rin
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