## Bulletin of the Belgian Mathematical Society - Simon Stevin

- Bull. Belg. Math. Soc. Simon Stevin
- Volume 25, Number 3 (2018), 467-480.

### On injectivity of the ring of real-valued continuous functions on a frame

Ali Akbar Estaji and Mostafa Abedi

#### Abstract

We give characterizations of $P$-frames and extremally disconnected $P$-frames based on ring-theoretic features of the ring of continuous real- valued functions on a frame $L$, i.e. $\mathcal RL$. It is shown that $L$ is a $P$-frame if and only if $\mathcal RL$ is an $\aleph_0$-self-injective ring. Consequently for pseudocompact frames if $\mathcal RL$ is $\aleph_0$-self-injective, then $L$ is finite. We also prove that $L$ is an extremally disconnected $P$-frame iff $\mathcal{R}L$ is a self-injective ring iff $\mathcal{R}L$ is a Baer regular ring iff $\mathcal{R}L$ is a continuous regular ring iff $\mathcal{R}L$ is a complete regular ring.

#### Article information

**Source**

Bull. Belg. Math. Soc. Simon Stevin, Volume 25, Number 3 (2018), 467-480.

**Dates**

First available in Project Euclid: 11 September 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.bbms/1536631239

**Digital Object Identifier**

doi:10.36045/bbms/1536631239

**Mathematical Reviews number (MathSciNet)**

MR3852680

**Zentralblatt MATH identifier**

06861554

**Subjects**

Primary: 06D22: Frames, locales {For topological questions see 54-XX} 16D50: Injective modules, self-injective rings [See also 16L60] 13A30: Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics 16D25: Ideals 54C30: Real-valued functions [See also 26-XX] 54G05: Extremally disconnected spaces, $F$-spaces, etc.

**Keywords**

$P$-frame Extremally disconnected frame self-injective $\aleph_0$-self-injective Ring of continuous real-valued functions on a frame

#### Citation

Estaji, Ali Akbar; Abedi, Mostafa. On injectivity of the ring of real-valued continuous functions on a frame. Bull. Belg. Math. Soc. Simon Stevin 25 (2018), no. 3, 467--480. doi:10.36045/bbms/1536631239. https://projecteuclid.org/euclid.bbms/1536631239