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.