This paper was inspired by the definition and study of injectivity domains (as in \textit{A.~N. Alahmadi, M. Alkan} and \textit{S. López-Permouth}, [Glasg. Math. J. 52A, 7-17 (2010; Zbl 1228.16004)] and \textit{N. Er, S. López-Permouth} and \textit{N. Sökmez} [J. Algebra 330, No. 1, 404-417 (2011; Zbl 1227.16004)]). The dual concepts are defined here: For any (unital right) \(R\)-module \(M\), the projectivity domain of \(M\) is defined as \(\{N\in\text{Mod-}R:M\) is \(N\)-projective\}. A module \(M\) is said to be projectively poor (p-poor) if its projectivity domain consists of precisely the semisimple \(R\)-modules. If a ring \(R\) is such that every \(R\)-module is either projective or projectively poor, then \(R\) is said to have no (right) p-middle class. Properties of p-poor modules are investigated first. For example, it is shown that if \(M_R\) is a p-poor module, then so is \(M\oplus N\) for all \(N\in\text{Mod-}R\). Restrictions are then placed on the ring. Semisimple Artinian rings are characterized in terms of p-poor \(R\)-modules. It is proved that every ring has a semisimple p-poor module. If \(R\) is semilocal, then \((R/J(R))_R\) is shown to be p-poor and if \(R\) is a right PCI-domain, then \(E(R)\) (and consequently every nonzero injective module) is p-poor. Rings with no (right) p-middle class are studied next. The following results are found for a ring \(R\) with no right p-middle class: (1) For any two-sided ideal \(I\), \(R/I\) is semisimple Artinian or \(I\) is a direct summand of \(R\). (2) If \(J(R)\neq 0\), then \(R\) is semilocal (3) \(\text{Soc}(R_R)\) is a direct summand of \(R\) or \(R\) is semiartinian with Loewy length \(\leq 2\). (4) If \(J(R)\neq J(R)^2\), then \(J(R)=0\). (5) Factor rings of \(R\) have no right p-middle class. (6) \(R\) is either semiprimary with \(J(R)^2=0\) or semiprime. (7) \(R\) does not contain an infinite independent family of nonzero two-sided ideals. (8) \(R\) does not contain an infinite set of central orthogonal idempotents. (9) If \(R\) is also indecomposable, then \(\text{Soc}(R_R)\leq_eR\) with \(J(R)^2=0\) or \(\text{Soc}(R_R)=0\). (10) If \(R\) is indecomposable semiprime, and \(R\) is not a prime ring, then \(R\) is semisimple Artinian. The main theorem states that every ring with no right p-middle class is the ring direct sum of a semisimple Artinian ring \(S\) and a ring \(K\) which is either zero or an indecomposable ring with exactly one of the following properties: (1) \(K\) is a semiprimary right SI-ring with \(J(K)\neq 0\), or (2) \(K\) is a semiprimary ring with \(\text{Soc}(K_K)=Z_r(K)=J(K)\neq 0\), or (3) \(K\) is a prime ring with \(\text{Soc}(K_K)=0\) and either \(J(K)=0\) or \(_KJ(K)\) and \(J(K)_K\) is infinitely generated, or (4) \(K\) is a prime right SI-ring with infinitely generated right socle. Some examples are found of rings that have no right p-middle class, including QF-rings with unique simple module (up to isomorphism) and \(J(R)^2=0\).