When rings of continuous functions are weakly regular. (Q2349218)

It is well known that, for any Tychonoff space \(X\), the ring \(C(X)\) is regular in the sense of von Neumann precisely when \(X\) is a \(P\)-space. This result also holds in the broader context of pointfree topology. Indeed, denoting the ring of real-valued continuous functions on a frame \(L\) by \(\mathcal RL\), then \(\mathcal RL\) is a regular ring iff \(L\) is a \(P\)-frame [\textit{B. Banaschewski} and \textit{S. S. Hong}, Commentat. Math. Univ. Carol. 44, No. 2, 245-259 (2003; Zbl 1098.06006)]. Another result of the type, due to \textit{T. Dube} and \textit{M. Matlabyana} [Topology Appl. 160, No. 12, 1345-1352 (2013; Zbl 1288.06019)], is that \(\mathcal RL\) is a quasi-regular ring (i.e., its classical ring of quotients is regular) iff \(L\) is cozero complemented. Similarly, \(\mathcal RL\) is an almost regular ring (i.e., each of its elements is either a zero-divisor or a unit) iff \(L\) is an almost \(P\)-frame [\textit{T. Dube}, Algebra Univers. 60, No. 2, 145-162 (2009; Zbl 1186.06006)]. Less restricted than almost \(P\)-frames are the weak almost \(P\)-frames introduced in the paper under review (extending the almost \(P\)-spaces of \textit{F. Azarpanah} and \textit{M. Karavan} [Czech. Math. J. 55, No. 2, 397-407 (2005; Zbl 1081.54013)]): a completely regular frame \(L\) is a \textit{weak almost \(P\)-frame} if whenever \(a\) and \(b\) are cozero elements of \(L\) with \(a^*\leq b^*\), then there is a dense cozero element \(c\) such that \(b\wedge c\leq a\). The main goal of the paper is to seek a ring-theoretic characterization of these frames. First, the authors introduce the following definition: a ring \(A\) is \textit{weakly regular} if for any \(a,b\in A\) with \(\mathrm{Ann}(a)\subseteq\mathrm{Ann}(b)\), there is a non-zero-divisor \(c\in A\) such that \(bc\in M(a)\) (where \(\mathrm{Ann}(x)\) denotes the annihilator of the element \(x\in A\) and \(M(x)\) is the intersection of all maximal ideals of \(A\) containing \(x\)). Then, in parallel with the results of Banaschewski-Hong, Dube-Matlabyana and Dube quoted at the beginning of this review, they show that a frame \(L\) is a weak almost \(P\)-frame iff \(\mathcal RL\) is a weakly regular ring. Several other results involving weak almost \(P\)-frames are presented. In the last section of the paper, some characterizations of weakly regular (reduced) \(f\)-rings are presented such as, for instance, that a reduced \(f\)-ring is weakly regular if and only if every prime \(z\)-ideal in it which contains only zero-divisors is a \(d\)-ideal. Putting all pieces together, the authors conclude the paper with the following nice diagram of implications depicting the position of weak regularity for (reduced) \(f\)-rings vis-à-vis other weaker variants of regularity: (1) Regularity \(\Longrightarrow\) almost regularity \(\Longrightarrow\) weak regularity. (2) Regularity \(\Longrightarrow\) quasi-regularity \(\Longrightarrow\) weak regularity. (3) Quasi-regularity + almost regularity \(\Longrightarrow\) regularity. This diagram is a perfect ring analogue of the diagram of irreversible implications that hold for the corresponding classes of frames: (1) \(P\)-frame \(\Longrightarrow\) almost \(P\)-frame \(\Longrightarrow\) weak almost \(P\)-frame. (2) \(P\)-frame \(\Longrightarrow\) cozero complemented \(\Longrightarrow\) weak almost \(P\)-frame. (3) Cozero complemented + almost \(P\)-frame \(\Longrightarrow\) \(P\)-frame.