Journal of Commutative Algebra

On subrings of the form $I+\mathbb {R}$ of $C(X)$

F. Azarpanah, M. Namdari, and A.R. Olfati

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In this article we observe that $C(X)$ is integral over $I+\mathbb R$ if and only if $I$ is of the form of a finite intersection of real maximal ideals. The integral closure of a subring $I+\mathbb R$ is investigated and it turns out that $I+\mathbb R$ is integrally closed if and only if $I$ is semiprime and $\theta (I)$ is connected. We show that $I^u+\mathbb R$, where $I^u$ is the uniform closure of the ideal $I$, is always a ring of quotients of $I+\mathbb R$ and $C(X)$ is a ring of quotients of $I+\mathbb R$ if and only if $I$ is either essential (large) or a maximal ideal generated by an idempotent. Maximal subrings of the form $I+\mathbb R$ of $C(X)$ are characterized and it is shown that $[M^p]^u+\mathbb R$ (which is isomorphic to $C(X\cup \{p\})$, where $X\cup \{p\}$ is a subspace of $\beta X$) is not contained in a proper maximal subring of $C(X)$. We have observed that for each subring of the form $I+\mathbb R$ of $C(X)$, the sum of every two prime ideals of $I+\mathbb R$ is prime or all of $I+\mathbb R$ if and only if $X$ is an $F$-space. Ideals $I$ are characterized for which $I+\mathbb R$ is a regular ring and for every $z$-ideal $I$, we have shown that the sum of every two $z$-ideals of $I+\mathbb R$ is a $z$-ideal or all of $I+\mathbb R$. Finally, some algebraic and topological properties are given for which $I+\mathbb R$ is a self-injective ring or a uem-ring.

Article information

J. Commut. Algebra, Volume 11, Number 4 (2019), 479-509.

First available in Project Euclid: 7 December 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 54C40: Algebraic properties of function spaces [See also 46J10]
Secondary: 13B22: Integral closure of rings and ideals [See also 13A35]; integrally closed rings, related rings (Japanese, etc.)

Integral closure valuation ring ring of quotients essential ideals maximal subring self-injective ring $F$-space


Azarpanah, F.; Namdari, M.; Olfati, A.R. On subrings of the form $I+\mathbb {R}$ of $C(X)$. J. Commut. Algebra 11 (2019), no. 4, 479--509. doi:10.1216/JCA-2019-11-4-479.

Export citation


  • M. Atiyah and I.G. MacDonald, Introduction to commutative algebra, Addison-Wesley, London (1969).
  • F. Azarpanah, Intersection of essential ideals in $C(X)$, Proc. Amer. Math. Soc. 125 (1997), 2149–2154.
  • F. Azarpanah and O.A.S. Karamzadeh, Algebraic characterizations of some disconnected spaces, Italian J. Pure Appl. Math. 12 (2002), 155–168.
  • F. Azarpanah and R. Mohamadian, $\sqrt{z}$-ideals and $\sqrt{z^\circ}$-ideals in $C(X)$, Acta Math. Sin. (Engl. Ser.) 23 (2007), 989–996.
  • F. Azarpanah, F. Manshoor and R. Mohamadian, Connectedness and compactness in $C(X)$ with the $m$-topology, Topology Appl. 159 (2012), 3486–3493.
  • F. Azarpanah and A.R. Olfati, On ideals of ideals in $C(X)$, Bull. Iran. Math. Soc. 41 (2015), 23–41.
  • D.E. Dobbs, Every commutative ring has a minimal ring extension, Comm. Algebra 34 (2006), 3875–3881.
  • J.M. Dominguez and M.A. Mulero, Finite homomorphisms on rings of continuous functions, Topology Appl. 137 (2004), 115–124. \vfill
  • R. Engelking, General topology, Heldermann Verlag, Berlin (1989). \vfill
  • D. Ferrand and J.P. Oliver, Homomorphisms minimaux d'anneaux, J. Algebra 16 (1970), 461–471. \vfill
  • N.J. Fine, L. Gillman and J. Lambek, Rings of quotients of rings of functions, McGill Univ. Press, Montréal, Quebec (1965). \vfill
  • L. Fuchs and L. Salce, Modules over non-Noetherian domains, Mathematical Surveys and Monographs 84, Amer. Math. Soc., Providence, RI (2001). \vfill
  • L. Gillman and M. Jerison, Rings of continuous functions, Springer (1976). \vfill
  • D.G. Johnson and M. Mandelker, Functions with pseudocompact support, General Topology Appl. 3 (1973), 331–338. \vfill
  • O.A.S. Karamzadeh and M. Rostami, On the intrinsic topology and some related ideals of $C(X)$, Proc. Amer. Math. Soc. 93 (1985), 179–184. \vfill
  • O.A.S. Karamzadeh, M. Motamedi and S.M. Shahrtash, On rings with a unique proper essential right ideal, Fund. Math. 183 (2004), 229–244. \vfill
  • G. Mason, Prime $z$-ideals in $C(X)$ and related rings, Canad. Math. Bull. 23 (1980), 437–443. \vfill
  • S. Mrowka, Continuous functions on countable subspaces, Port. Math. 29 (1970), 177–180. \vfill
  • M.A. Mulero, Algebraic properties of rings of continuous functions, Fund. Math. 149 (1996), 55–66. \vfill
  • P. Nanzetta and D. Plank, Closed ideals in $C(X)$, Proc. Amer. Math. Soc. 35 (1972), 601–606. \vfill
  • M.C. Rayburn, Compactifications with almost locally compact outgrowth, Proc. Amer. Math. Soc. 106 (1989), 223–229. \vfill
  • D. Rudd, On two sum theorems for ideals of $C(X)$, Michigan Math. J. 17 (1970), 139–141. \vfill
  • D. Rudd, On isomorphisms between ideals in rings of continuous functions, Trans. Amer. Math. Soc. 159 (1971), 335–350. \vfill
  • D. Rudd, An example of a $\Phi$-algebra whose uniform closure is a ring of continuous functions, Fund. Math. 77 (1972), 1–4. \vfill
  • D. Rudd, On structure spaces of ideals in rings of continuous functions, Trans. Amer. Math. Soc. 190 (1974), 393–403. \vfill
  • D. Rudd, $P$-ideals and $F$-ideals in rings of continuous functions, Fund. Math. 88 (1975), 53–59. \vfill
  • E.M. Vechtomov, The lattice of subalgebras of the ring of continuous functions and Hewitt spaces, Mat. Zametki 62 (1997), 687–693. In Russian; translated in Math. Notes 62 (1997), 575–580.