Commutative ring extensions defined by perfect-like conditions - Alaoui Ismaili, K.; Mahdou, N.; Moutui, M. A. S.; Alaoui Ismaili, K.; Mahdou, N.; Moutui, M. A. S. (Institute of Mathematics, NAS of Ukraine, 2023)
UDC 512.5 In 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.