@article{Alaoui Ismaili_Mahdou_Moutui_2023, title={Commutative ring extensions defined by perfect-like conditions}, volume={75}, url={https://umj.imath.kiev.ua/index.php/umj/article/view/6878}, DOI={10.37863/umzh.v75i3.6878}, abstractNote={<p>UDC 512.5</p> <p>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 $n$.<span class="Apple-converted-space"> </span>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<span class="Apple-converted-space"> </span>projective dimension less or equal than $n$.<span class="Apple-converted-space"> </span>Recently Purkait defined a ring $R$ to be $n$-semiperfect if $\overline{R}=R/{\rm Rad}(R)$ is semisimple and $n$-potents lift modulo ${\rm Rad}(R).$<span class="Apple-converted-space"> </span>We study<span class="Apple-converted-space"> </span>of three classes of rings, namely, $n$-perfect, strongly $n$-perfect, and $n$-semiperfect rings.<span class="Apple-converted-space"> </span>We investigate these notions in several ring-theoretic structures with an aim of construction of new original families of examples satisfying the<span class="Apple-converted-space"> </span>indicated properties and subject to various ring-theoretic properties.</p>}, number={3}, journal={Ukrains’kyi Matematychnyi Zhurnal}, author={Alaoui Ismaili, K. and Mahdou, N. and Moutui, M. A. S.}, year={2023}, month={Apr.}, pages={319-327} }