Commutative ring extensions defined by perfect-like conditions

K.  Alaoui Ismaili; N.  Mahdou; M. A. S.  Moutui

doi:10.37863/umzh.v75i3.6878

K. Alaoui Ismaili Laboratory of Mathematics, Computing and Applications-Information Security (LabMia-SI), Department of Mathematics, Faculty of Sciences of Rabat, Mohammed V University in Rabat, Morocco
N. Mahdou Department of Mathematics, Faculty of Science and Technology of Fez, University S. M. Ben Abdellah Fez, Morocco
M. A. S. Moutui University of Haute Alsace, IRIMAS, De ́partement de Mathe ́matiques, Mulhouse, France and Division of Science, Technology and Mathematics, American University of Afghanistan, Doha Campus, Qatar

DOI: https://doi.org/10.37863/umzh.v75i3.6878

Keywords: perfect, n-perfect, strongly n-perfect, (n, d)-ring, nsemiperfect, Von Neumann Regular ring, amalgamated algebra, amalgamated duplication.

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 $\overline{R}=R/{\rm Rad}(R)$ 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.

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