Journal of Commutative Algebra

When is $C(X)$ polynomially ideal?

Karim Boulabiar and Samir Smiti

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Abstract

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.

Article information

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

Dates
First available in Project Euclid: 19 January 2016

Permanent link to this document
https://projecteuclid.org/euclid.jca/1453211670

Digital Object Identifier
doi:10.1216/JCA-2015-7-4-473

Mathematical Reviews number (MathSciNet)
MR3451352

Zentralblatt MATH identifier
1352.13002

Subjects
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}

Keywords
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

Citation

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. https://projecteuclid.org/euclid.jca/1453211670


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References

  • F. Azarpanah, When is $C(X)$ a clean ring?, Acta Math. Hung. 94 (2002), 53–58.
  • B.A. Barnes, Algebraic elements of a Banach algebra modulo an ideal, Pac. J. Math. 117 (1985), 219–231.
  • S.J. Bernau and C.B. Huijsmans, Almost $f$-algebras and $d$-algebras, Math. Proc. Cambr. Philos. Soc. 107 (1990), 287–308.
  • F. Beukers, C.B. Huijsmans and B. de Pagter, Unital embedding and complexifications of $f$-algebras, Math. Z. 18 (1983), 131–144.
  • A. Bigard, K. Keimel and S. Wolfenstein, Groupes et Anneaux Réticulés, Lect. Notes Math. 608, Springer-Verlag, Berlin, 1977.
  • G. Birkhoff and R.S. Pierce, Lattice-ordered rings, An. Acad. Brasil. Ci. 28 (1956), 41–69.
  • K. Boulabiar, G. Buskes and G. Sirotkin, Algebraic order bounded disjointness preserving operators and strongly diagonal operators, Int. Equat. Oper. Theor. 54 (2006), 9–31.
  • L. Gillman, M. Henriksen and M. Jerison, On a theorem of Gelfand and Kolmogoroff concerning maximal ideals in rings of continuous functions, Proc. Amer. Math. Soc. 5 (1954), 447–455.
  • L. Gillman and M. Jerison, Rings of continuous functions, Springer, Berlin, 1976.
  • M. Henriksen, J.R. Isbell and D.G. Johnson, Residue class fields of lattice-ordered algebras, Fund. Math. 50 (1961), 107–117.
  • E. Hewitt, Rings of real-valued continuous functions I, Trans. Amer. Math. Soc. 64 (1948), 45–99.
  • W.A.J. Luxemburg and A.C. Zaanen, Riesz spaces I, North-Holland, Amsterdam, 1971.
  • W.W. McGovern, Clean semiprime $f$-rings with bounded inversion, Comm. Alg. 31 (2003), 3295–3304.
  • C.R. Miers, Polynomially ideal $C^{\ast}$-algebras, Amer. Math. J. 98 (1976), 165–170.
  • W.K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc. 229 (1977), 269–278.
  • C.L. Olsen, A structure theorem for polynomially compact operator, Amer. J. Math. 93 (1971), 686–698.
  • S.A. Steinberg, Lattice-ordered rings and modules, Springer, New York, 2010.
  • A.C. Zaanen, Introduction to operator theory in Riesz spaces, Springer, Berlin, 1997.
  • ––––, Riesz spaces II, North-Holland, Amsterdam, 1983.