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Winter 2020 On the sum of $z$-ideals in subrings of $C(X)$
Fariborz Azarpanah, Mehdi Parsinia
J. Commut. Algebra 12(4): 459-466 (Winter 2020). DOI: 10.1216/jca.2020.12.459

Abstract

Let X be a Tychonoff space and A(X) be an intermediate subalgebra of C(X), i.e., C(X)A(X)C(X). We show that such subrings are precisely absolutely convex subalgebras of C(X). An ideal I in A(X) is said to be a zA-ideal if Z(f)Z(g), fI and gA(X) imply that gI. We observe that the coincidence of zA-ideals and z-ideals of A(X) is equivalent to the equality A(X)=C(X). This shows that every z-ideal in A(X) need not be a zA-ideal and this is a point which is not considered by D. Rudd in Theorem 4.1 of Michigan Math. J. 17 (1970), 139–141, or by G. Mason in Theorem 3.3 and Proposition 3.5 of Canad. Math. Bull. 23:4 (1980), 437-443. We rectify the induced misconceptions by showing that the sum of z-ideals in A(X) is indeed a z-ideal in A(X). Next, by studying the sum of z-ideals in subrings of the form I+ of C(X), where I is an ideal in C(X), we investigate a wide class of examples of subrings of C(X) in which the sum of z-ideals need not be a z-ideal. It is observed that, for every ideal I in C(X), the sum of any two z-ideals in I+ is a z-ideal in I+ or all of I+ if and only if X is an F-space. This result answers a question raised by Azarpanah, Namdari and Olfati in J. Commut. Algebra 11:4 (2019), 479–509.

Citation

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Fariborz Azarpanah. Mehdi Parsinia. "On the sum of $z$-ideals in subrings of $C(X)$." J. Commut. Algebra 12 (4) 459 - 466, Winter 2020. https://doi.org/10.1216/jca.2020.12.459

Information

Received: 18 November 2017; Revised: 16 May 2018; Accepted: 28 May 2018; Published: Winter 2020
First available in Project Euclid: 5 January 2021

MathSciNet: MR4194935
Digital Object Identifier: 10.1216/jca.2020.12.459

Subjects:
Primary: 46E25, ‎54C30

Rights: Copyright © 2020 Rocky Mountain Mathematics Consortium

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Vol.12 • No. 4 • Winter 2020
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