Journal of Commutative Algebra

When is $C(X)$ polynomially ideal?

Karim Boulabiar and Samir Smiti

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Let $\mathbb{A}$ be a commutative $f$-algebra with unit. The sets of all ideals in $\mathbb{A}$ and all intersections of maximal ideals in $\mathbb{A}$ are denoted by $\mathfrak{I}(\mathbb{A})$ and $\mathfrak{IM}% (\mathbb{A})$, respectively. Whenever $\mathfrak{a}% \in\mathfrak{I}(\mathbb{A})$, we say that $\mathbb{A}$ is polynomially $\mathfrak{a}$-ideal if, for every $f\in\mathbb{A}$ with $p(f)\in\mathfrak{a}$ for some non-zero polynomial $p(x)$, there is an $f_{0}\in\mathfrak{a}$ such that $p(f+f_{0})=0$. We prove that if $\mathbb{A}$ is bounded inversion closed and $\mathfrak{a}\in\mathfrak{IM}(\mathbb{A})$, then $\mathbb{A}$ is polynomially $\mathfrak{a}$-ideal if and only if idempotents lift modulo $\mathfrak{a}$. This fact is based upon a systematic study of idempotent elements of an $f$-algebra. As a consequence, we show that, if $X$ is a Tychonoff space, then $C(X)$ is polynomially $\mathfrak{a}$-ideal for all $\mathfrak{a}\in\mathfrak{I}(C(X))$ if and only if $X$ is a $P$-space. Moreover, we prove that $C(X)$ is polynomially $\mathfrak{a}$-ideal for all $\mathfrak{a}\in\mathfrak{IM}(C(X))$ if and only if $X$ is strongly zero-dimensional. It turns out that this extends a theorem by Miers, namely, if $X$ is a compact Hausdorff space, then $C(X)$ is polynomially $\mathfrak{a}$-ideal for every uniformly closed ideal $\mathfrak{a}$ in $C(X)$ if and only if $X$ is totally disconnected.

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J. Commut. Algebra, Volume 7, Number 4 (2015), 473-493.

First available in Project Euclid: 19 January 2016

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Primary: 06F25: Ordered rings, algebras, modules {For ordered fields, see 12J15; see also 13J25, 16W80} 13A15: Ideals; multiplicative ideal theory 46E25: Rings and algebras of continuous, differentiable or analytic functions {For Banach function algebras, see 46J10, 46J15}

Commutative f-algebra idempotent element idempotent lift modulo an ideal polynomially ideal bounded inversion closed characteristic function intersection of maximal ideals completely regular space lattice-ordered algebra


Boulabiar, Karim; Smiti, Samir. When is $C(X)$ polynomially ideal?. J. Commut. Algebra 7 (2015), no. 4, 473--493. doi:10.1216/JCA-2015-7-4-473.

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