Commutative ring extensions defined by perfect-like conditions - Alaoui Ismaili, K.; Mahdou, N.; Moutui, M. A. S. (Ukrains’kyi Matematychnyi Zhurnal, 2023)
UDC 512.5In 2005, Enochs, Jenda, and López-Romos extended the notion of perfect rings to $n$-perfect rings such that a ring is $n$-perfect if every flat module has projective dimension less or equal than $n$.  Later, Jhilal and Mahdou defined a commutative unital ring $R$ to be strongly $n$-perfect if any $R$-module of flat dimension less or equal than $n$ has a  projective dimension less or equal than $n$.  Recently Purkait defined a ring $R$ to be $n$-semiperfect if $\overline{R}=R/{\rm Rad}(R)$ is semisimple and $n$-potents lift modulo ${\rm Rad}(R).$  We study  of three classes of rings, namely, $n$-perfect, strongly $n$-perfect, and $n$-semiperfect rings.  We investigate these notions in several ring-theoretic structures with an aim of construction of new original families of examples satisfying the  indicated properties and subject to various ring-theoretic properties.