Commutative ring extensions defined by perfect-like conditions

Abstract

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 R ¯ = R / R a d ( R ) is semisimple and n -potents lift modulo R a d ( 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.