## Journal of Commutative Algebra

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

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

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

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 7 December 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.jca/1575687731

**Digital Object Identifier**

doi:10.1216/JCA-2019-11-4-479

**Mathematical Reviews number (MathSciNet)**

MR4039979

**Zentralblatt MATH identifier**

07147393

**Subjects**

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.)

**Keywords**

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

#### Citation

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. https://projecteuclid.org/euclid.jca/1575687731